Scuola di Dottorato in Ingegneria Civile
Tesi di Dottorato:
STATICS AWARE VORONOI GRID-SHELLS
Dottorando:
Ing. Davide Tonelli
Relatore:
Prof. Ing. Maurizio Froli
Submitted to the Department of Civil Engineering D.I.C.I. for the Degree of
Doctor of Philosophy in Civil Engineering
at the
University of Pisa
completed 30 January 2015
© 2015 University of Pisa.
All rights reserved.
Certified by:
Chris J. K. Williams Senior Lecturer of Architecture & Civil Engineering University of Bath Host
Accepted by:
Stefano Pagliara Associate Professor of Hydraulics University of Pisa Chair of the Doctoral Program
Accepted by:
Stefano Bennati Professor of Solid and Structural Mechanics University of Pisa Chair of the Doctoral School of Engineering
Abstract
This dissertation presents the Statics Aware Voronoi remeshing, a novel pattern for generating innovative free-form grid-shells that are both aesthetically pleasing and structurally sound. Two main novelties are introduced in the grid-shell design context: a polygonal topology and an automatic method for taking into account the statics of the surface underlying the grid-shell. These features, together with the innate adaptivity of the Voronoi pattern, make this new remeshing extremely suitable for free-form architecture.
Numerical analyses are carried out to show that free-form Statics Aware Voronoi grid-shells achieve better structural performances with respect to their quadrilateral competitors. Ad-ditionally, numerical results are confirmed and supported through a theoretical framework, instituted upon the old-time concept of equivalent continuum. Eventually, a physically built mock-up practically demonstrates the feasibility of the new Statics Aware Voronoi Grid-Shells and caters for the calibration of numerical models.
Grid-shells, topology, connectivity, Voronoi, hex-dominant, polygonal, metric, geometry, free-form, sur-face, skin, limit analysis, geometrically non-linear analysis, imperfection sensitivity, buckling, regular tilings, equivalent stiffness, equivalent thickness, mock-up, incremental load test.
1.2.3 Ruled Surfaces . . . 20 1.2.4 Hexagonal Topology . . . 21 2 Differential Geometry 27 2.1 Introduction. . . 27 2.2 Regular Curves . . . 27 Maps . . . 27 Regular Curves . . . 28
Tangent vector of a Regular Curve . . . 28
Curvature of Regular Curves . . . 28
Curvature of Planar Regular Curves in form of Graphs . . . 28
Torsion of Regular Curves . . . 28
Frenet Trihedron and Terminology . . . 29
2.3 Regular Surfaces . . . 29
Chain Rule. . . 29
Differential of a Map at a Point . . . 30
Chain Rule for Maps . . . 30
Regular Surfaces. . . 31
Regular Surfaces in form of Graphs . . . 32
Relation between Maps and Graphs . . . 32
Differentiable Mappings between Surfaces . . . 32
Tangent Plane . . . 32
First Fundamental Form . . . 33
Anisotropy Ellipse. . . 34
Gauss Map. . . 35
Differential of the Gauss Map . . . 35
Second Fundamental Form . . . 35
Meusnier Theorem . . . 35
Principal Curvatures and Principal Directions . . . 36
Curvature Tensor . . . 37
Lines of Curvature . . . 37
Asymptotic Curves . . . 37
Gaussian and Mean Curvatures . . . 37
Classification of Surface’s Points . . . 37
Conjugate Directions . . . 38
Differential of the Gauss Map and Second Fund.l Form in Local Coord. . . 38
Principal Curvatures in Local Coordinates . . . 39
Lines of Curvature in Local Coordinates . . . 39
Asymptotic Curves in Local Coordinates . . . 40
Specialization for Surfaces in form of Graphs. . . 40
Isometries . . . 40
Conformal Maps. . . 41
Vector Fields. . . 41
Trajectories on Vector Fields. . . 41
Vector Fields on Regular Surfaces . . . 41
Christoffel Symbols and Theorema Egregium. . . 42
Covariant Derivative . . . 42
Parallel Vector Fields . . . 43
Parallel Transport. . . 43
Geodesic Curves. . . 43
Geodesic Curvature . . . 44
Homeomorphism . . . 44
Euler-Poincar´e Characteristic and Genus of Surfaces . . . 44
Integral of a Function on a Surface . . . 45
Gauss Theorem . . . 45
Gauss-Bonnet and Cohen-Vossen Theorems . . . 46
2.4 Differential Operators on Surfaces . . . 46
Manifold . . . 46
Riemannian Manifolds . . . 47
Riemannian 2-Manifolds . . . 47
Riemannian Metric . . . 47
Riemannian Metric in Local Coordinates . . . 47
Riemannian 2-Manifolds in Local Coordinates . . . 48
Divergence on Surfaces . . . 48
Gradient on Surfaces . . . 48
Laplace-Beltrami Operator . . . 49
3 Geometry Processing 51 3.1 Geometry Processing Overview . . . 51
3.2 Surface Representation. . . 52
3.2.1 Parametric Surface Representation . . . 52
3.2.2 Implicit Surface Representation . . . 53
3.2.3 Surface Manifoldness. . . 53
3.2.4 Surface Smoothness . . . 53
3.3 Discrete Differential Geometry . . . 53
3.3.1 Surface Normal . . . 54 3.3.2 Surface Gradient . . . 54 3.3.3 Laplace-Beltrami Operator . . . 55 3.3.4 Surface Curvatures . . . 57 3.4 Surface Smoothing . . . 58 3.4.1 Diffusion Flow . . . 58 3.4.2 Fairing. . . 59 3.5 Surface Parameterization . . . 60 3.6 Surface Remeshing . . . 62
3.6.1 Mesh Global Structure. . . 62
3.6.2 Voronoi Diagrams . . . 63
3.6.3 Triangular Remeshing . . . 63
3.6.4 Quadrilateral Remeshing . . . 66
4.4.2 Elastic Strain Energy Derivation . . . 84
4.4.3 Virtual Work Derivation. . . 89
II
Statics Aware Voronoi Remeshing
91
5 Statics Aware Voronoi Grid-Shells: Geometry Generation 93 5.1 Voronoi-like Topology . . . 955.2 Statics Awareness. . . 97
5.2.1 Linear Static Analysis of the Surface as a Continuous Shell . . . 98
5.2.2 From Shell to Grid-Shell: Local Conversion Criteria . . . 98
5.3 Remeshing. . . 105
5.3.1 Regularization . . . 105
5.3.2 Smoothing and Symmetrization. . . 108
5.3.3 Tuning anisotropy and density . . . 109
5.3.4 Results and comparisons. . . 110
5.3.5 Influence of user parameters (D, A). . . 114
6 Statics Aware Voronoi Grid-Shells: Structural Analysis 117 6.1 Stability checks for grid-shells . . . 118
6.1.1 Analysis of imperfection sensitivity . . . 119
6.2 Experimental setup. . . 121
6.2.1 Datasets. . . 121
6.2.2 Restraints, load conditions, numerical modeling. . . 124
6.2.3 Statics comparison criterion . . . 125
6.3 Results. . . 125
6.3.1 Comparative imperfection sensitivity analysis . . . 126
6.3.2 Comparative analysis of ‘worst’ response diag. VS remeshing . . . 127
6.3.3 Response diagram VS imperfection amplitude. . . 129
6.4 Summary of Results . . . 129
7 Elastic Properties of Regular Tilings 131 7.1 Tiling Stiffness Parameters . . . 132
7.2 Plane Stress . . . 133
7.2.1 Plane Stress in Isotropic Materials . . . 133
7.2.2 Plane Stress in Orthotropic Materials . . . 133
7.3 Coordinates for Stress Tensor . . . 134
7.3.1 Coordinate Change for Stress Tensor . . . 135
7.4 Strain Energy . . . 135
7.4.1 Bidimensional Bodies under Plane Stress . . . 135
7.4.2 Beams . . . 135
7.5 Isotropic Triangular Pattern . . . 136
7.5.1 Equivalent Membrane Stiffness . . . 136
7.5.2 Equivalent Bending Stiffness . . . 136
7.5.3 Equivalent Thickness. . . 138
7.6.1 Equivalent Membrane Stiffness . . . 139
7.6.2 Equivalent Bending Stiffness . . . 142
7.6.3 Equivalent Thickness. . . 144
7.7 Isotropic Hexagonal Pattern. . . 144
7.7.1 Equivalent Membrane Stiffness . . . 144
7.7.2 Equivalent Bending Stiffness . . . 147
7.7.3 Equivalent Thickness. . . 149
7.8 Topologies Stiffnesses in Comparison . . . 149
7.9 Conclusions . . . 151
8 Statics Aware Voronoi Grid-Shells: Mock-up 153 8.1 The Digital Handling of the Mock-up. . . 154
8.1.1 Automated parametric F.E.M. analysis . . . 155
8.1.2 Joints Design and Production . . . 155
8.1.3 Beams Handling . . . 157
8.1.4 Panels Design and Production . . . 157
8.2 Assembling . . . 158
8.3 The Structural Behaviour of the Mock-up . . . 162
8.3.1 Experimental Incremental Load Tests . . . 162
8.3.2 Numerical Incremental Load Tests . . . 166
9 Conclusions 171 9.1 Summary of Results . . . 171
9.1.1 Stiffness of the Hexagonal Topology . . . 171
9.1.2 Competitiveness of Hex-dominant and Voronoi-like Grid-Shells . . . 172
1.9 (a) Hamburg History Museum Court Roof, Hamburg, completed in 1989. (b) Hyppo
House, Berlin Zoo, completed in 1996. . . 10
1.10 (a) DG Bank Court Roof, Berlin, completed in 1998. (b) Maritime Museum, Osaka, completed in 2000. . . 10
1.11 British Museum Great Court Roof, London, completed in 2000. . . 10
1.12 (a) BMW Welt, Monaco, completed in 2007. (b) Blob, Eindhoven, completed in 2010. 11 1.13 (a) The Gherkin, London, completed in 2003. (b) Aldar Headquarters, Abu Dhabi, completed in 2010. (c) Capital Gate, Abu Dhabi, completed in 2011. . . 11
1.14 (a) Guggenheim Museum, Bilbao, completed in 1997. (b) Foundation Louis Vouitton, Paris, completed in 2012. . . 11
1.15 (a) Ne¨umunster Abbey Court Roof, Luxembourg, completed in 2006. (b) Strasbourg Train Station Glass Facade, Strasbourg, completed in 2008. . . 12
1.16 (a) Cabot Circus Grid-Shell, Bristol, completed in 2008. (b) Yas Hotel Grid-Shell, Abu Dhabi, completed in 2009. . . 12
1.17 (a) Vela Fiera Milano-Rho, Milano, completed in 2005. (b) Islamic Art Pavilion, Louvre Paris, completed in 2012. . . 12
1.18 (a) Smithsonian Institution Court Roof, Washington DC, completed in 2007. (b) Zlote Tarasy Shopping Centre, Warsaw, completed in 2007. . . 13
1.19 Gardens by the Bay, Singapore, completed in 2014. . . 13
1.20 (a) Eden Project, Cornwall, completed in 2000. (b) Kreod Pavilion, London (temporary installation), completed around 2011.. . . 13
1.21 (a) Nine Bridges Golf House, Seoul, completed in 2010. (b) Centre Pompidou, Metz, completed in 2010. . . 14
1.22 (a) Rokko Shidare Observatory, Kobe, completed in 2013. . . 14
1.23 (a) Catalyst anisotropic hexagonal corrugated thin-shell, Minneapolis, mock-up com-pleted in 2012. . . 14
1.24 Mutsuro Sasaki masterpieces. (a) Kakamigahara Crematorium, Gifu 2006. (b) Rolex Learning Centre, Lausanne, 2010. (c) Teshima Art Museum, Kagawa, 2010. . . 15
1.25 (a) Buckminster Fuller geodesic dome in Montreal, completed in 1967. (b) Local refine-ment of a triangular grid topology. . . 16
1.26 (a) Geometric Torsion in a Triangular Grid. (b) Geometric Torsion in a Node of the project “La Vela - Milano Fiere” (see Figure 1.17(a)), Milano (IT). . . 17
1.27 Two extremal orientations of the isotropic quadrilateral grid. . . 17
1.28 Scale-Trans Meshes [GSC+04], [Sch05].. . . 18
1.29 PQ perturbation algorithm and conical meshes [LPW+06].. . . 19
1.30 Geometrical construction of the conjugacy relationship at a point P of the surface Φ, respectively hyperbolic, elliptic and parabolic [ZSW10]. . . 20
1.31 Developable strips [PSB+08]. . . . . 20
1.32 Paneling algorithm [EKS+10].. . . . 21
1.33 Design for the Cagliari contemporary art museum. Paneling of certain portions of the skin through ruled surfaces [FP10]. . . 21
1.34 Honeycomb subdivision algorithm [AS03]. . . 23
1.35 Heuristic planar hex-dominant remeshing algorithm [CW07]. . . 23
1.36 TPI algorithm [Tro08]. . . 24
1.37 Edge offset meshes [PLW+07].. . . 25
1.38 CP meshes [SHWP09]. . . 25
1.39 Free-form honeycomb structures [JWWP14]. . . 26
2.1 Curvature K at point M (x, y) [Pis65, p. 218]. . . 29
2.2 Frenet trihedron at s [dC76, p. 17]. . . 30
2.3 Differential of a map at a point p [dC76, p. 127]. . . 31
2.4 Regular surface [dC76, p. 54]. . . 32
2.5 Differentiable mapping between surfaces [dC76, p. 73]. . . 33
2.6 Tangent plane and tangent vector to a regular surface S [dC76, p. 84]. . . 33
2.7 Anisotropy Ellipse in a point p of the surface S [HLS07].. . . 35
2.8 Normal curvature [dC76, p. 141]. . . 36
2.9 Conjugacy relationship at an elliptic point: geometric construction [dC76, p. 150]. . . 38
2.10 Differentiable vector field and trajectories [dC76, p. 175].. . . 41
2.11 Physical meaning of the Covariant Derivative linear operator [dC76, p. 238]. . . 43
2.12 Geodesic Curve [dC76, p. 249]. . . 44
2.13 Euler-Poincar´e characteristic for some simple compact surfaces [dC76, p. 273].. . . 45
2.14 Geodesic triangle T , respectively on a pseudosphere (K = −1, left) and on a sphere (K = 1, right) [dC76, p. 265]. . . 46
2.15 Differentiable manifold: ‘a set provided with a differentiable structure’ [dC92, p. 3].. . 47
3.1 Example of Catmull-Clark subdivision scheme [Wik14h]. . . 52
3.2 Examples of non-manifold surfaces, respectively non-manifold vertex (left) and edge (right) [BKP+10].. . . 53
3.3 Computation of per-vertex area: respectively a barycentric cell (left) and a Voronoi cell (right) [BKP+10].. . . . 54
3.4 Barycentric basis functions (also referred to as shape functions in the F.E. context (see section 4.4.2)) [BKP+10]. . . 55
3.5 Averaging area for the derivation of the discrete Laplace-Beltrami and discrete Gaussian Curvature operators [BKP+10]. . . 56
3.6 Mesh smoothing obtained through the diffusion flow technique [DMSB99]. . . 59
3.7 Results of fairness functionals minimization, respectively membrane surface (left), thin plate surface (center) and minimum variation of curvature surface (right) [BK04b]. . . 60
3.8 Different parametrizations of a complex 3D surface cut in patches homeomorphic to a disk (left) [SH02], respectively obtained through a barycentric mapping (center) [BKP+10] and a conformal free boundary mapping (right) [HLS07].. . . 61
3.9 Triangular mesh remeshings: irregular (left), semi-regular (center), regular (right) [BKP+10]. 62 3.10 Voronoi Diagram of a 2D point set (thick line) and its dual Delaunay Triangulation (dotted line). . . 63
3.11 Results of the Greedy algorithms for isotropic triangular remeshing [DR10]. . . 64
3.12 Results of the Incremental algorithms for isotropic triangular remeshing [BK04a]. . . . 64
3.13 Centroidal Voronoi tessellation with constant density function ρ(x) [DFG99]. . . 65
3.14 Centroidal Voronoi tessellation: different results obtained by simply varying the density function ρ(x) [DFG99].. . . 65
3.15 Uniform remeshing through centroidal Voronoi tessellation: initial sampling in 2D pa-rameter domain (top left), uniform point sampling in 2D papa-rameter domain (top right), centroidal Voronoi tessellation (bottom left), uniform triangular remeshing with feature preservation (bottom center and right) [BKP+10]. . . 66
point), F (failure point). . . 78
4.5 Explicit (left), implicit (center) and mixed (right) F.E.M. analyses. . . 78
4.6 2D beam shape functions: see this Figure and equation (4.42). . . 85
4.7 Beam finite displacement. . . 87
5.1 Four examples of internally statically determinate cells with different topologies: small circles on the vertices stand for cylindrical hinges, whereas small triangles next to the vertices mean flexural continuity. . . 94
5.2 Planarization of polygonal faces. An application of the quad-based algorithm proposed in [BDS+12] leads to flat but concave faces (left). A trade-off between planarity and regularity of faces obtained with the regularization technique described in section 5.3.1 (right).. . . 94
5.3 Voronoi-like and centroidal Voronoi-like patterns in nature. . . 96
5.4 A purely hexagonal mesh (left) does not fit in with all the boundary constraints, whereas a centroidal Voronoi tessellation (right) does. . . 97
5.5 Problem statement. Local fitting of a regular hexagonal grid in a neighbourhood Πp of a point p ∈ S, aligned with the principal directions ~uf, ~vf and subject to boundary forces | ~uf|, | ~vf| (left). Geometric dimensions of isotropic (top right) and anisotropic cells (bottom right) in Πp.. . . 99
5.6 Derivation of Udfor an isotropic hex-grid subject to an isotropic state of stress. . . . 101
5.7 From top left to bottom right: rhombus, rectangle, anisotropic hexagon and ellipse. Four reference geometries for computing Ua(β,hl). . . 102
5.8 Derivation of the relationship min h Ua( ~ vf ~ uf, b a) i for an anisotropic hex-grid subject to an anisotropic state of stress. . . 103
5.9 ‘Transfer’ Criteria: a new metric gΨ over S. . . 104
5.10 Density, anisotropy and directional field of the British dataset (top row) (see Table 5.1 and Figure 5.17 for dataset definition); the resulting deformed domain mesh and the corresponding undeformed domain with the resulting anisotropic, density-varying distance field for a set of samples on the surface (bottom row). . . 106
5.11 The guiding field (density, anisotropy and line field) for the Neumunester dataset (top); hex dominant ACVT tessellation obtained using [VC04] and coloured diagram depicting closeness of mesh faces to prescribed density and anisotropy values on a blue-to-red error scale (bottom). . . 107
5.12 A single face f of the ACVT remeshing, with the eigenvectors resulting from PCA (left); the un-stretched polygon f0with the aligned target polygon pt(f0) (middle); the computed displacement vectors in the original space (right). . . 107
5.13 ACVT remeshing (left), optimization for planarity using Shape-Up method [BDS+12] (middle) and optimization towards polygon regularity using the present method (right). The top row shows the effects on the planarity of the faces, whereas the bottom row displays the consequences on faces regularity. . . 108
5.14 Comparison of non-optimized versus symmetrized and optimized tessellation of the Shell dataset (see Figure 5.19 (top left) for the Shell dataset). . . 109
5.15 Smoothing the density (left), the anisotropy (middle), and the two orthogonal line fields (right) of the Botanic dataset: top row the original, bottom smoothed (see Table 5.1 and Figure 5.17 for dataset definition). . . 109
5.16 Symmetrization of Ψf for the Lilium dataset. Cross parametrization defined by two
symmetry planes (left); the density field (top row) and the line field (bottom row),
before and after symmetrization respectively (center and right).. . . 110
5.17 Comparison with state-of-the art remeshings, top views. From the top: Botanic, British and Lilium data-sets.. . . 112
5.18 Comparisons of distribution of Axial Forces on the datasets. Red corresponds to com-pression, blue corresponds to traction. . . 113
5.19 Some rendered examples of statics aware Voronoi remeshing : Shell dataset (top left), Lilium dataset [VHWP12] (top right), British Museum dataset [TSG+14] (bottom row).114 5.20 Influence of (D, A) parameters on λN Land δL, non-linear buckling multiplier and linear maximum displacement under serviceability load, respectively. Test on a 4×4 Shell model.115 6.1 Magnified deformed shapes for the statics aware Voronoi remeshing of the Neum¨unster dataset, side and front views. From top to bottom, respectively: ORIGINAL, LS, NLS, 1st LB eigenmode, 2nd LB eigenmode and 3rd LB eigenmode.. . . 120
6.2 The four dataset surfaces under consideration, both in top and perspective view. From top left to bottom right, respectively, the statics aware Voronoi remeshings of: Neum¨unster Abbey glass roof, British Museum Great Court Roof, Aquadom and Lilium Tower. The black bullet is the state parameter adopted in the geometrically non-linear analyses. . 122
6.3 Datasets of rather regular surfaces: Neum¨unster Abbey court roof (left) and British Mu-seum Great Court roof (right). From top to bottom, top views of: triangular, quadri-lateral and statics aware Voronoi remeshings. For each dataset, the statics equivalence criterion (see section 6.2.3) requires all the remeshings to have approximatively the same total length Ltot. See Table 6.1 for further details. . . 123
6.4 Datasets of totally free-form surfaces: Aquadom (left) and Lilium Tower top (right). From top to bottom, top views of: quadrilateral and statics aware Voronoi remeshings. For each dataset, the statics equivalence criterion (see section 6.2.3) requires all the remeshings to have approximatively the same total length Ltot. . . 124
6.5 Grid-spacing sensitivity analyses on a spherical cap (span-to-height ratio = 21.43). For this surface the triangular connectivity is very sensitive to grid-spacing variations. . . 125
6.6 All imperfection sensitivity results. From top left to bottom right: Neum¨unster Abbey, British Museum, Aquadom and Lilium Tower datasets. The horizontal lines represent the first linear buckling load computed on the perfect model. Text within graphs recalls the ‘worst’ geometric imperfection shape which was adopted in this analysis (see Section 6.1.1). . . 127
6.7 All the ‘worst’ response diagrams, respectively from top left to bottom right: Neum¨unster Abbey courtyard glass roof, British Museum great court roof, Aquadom and Lilium Tower top. The horizontal lines represent the ‘safety’ unit load factor. Text within the graphs recalls the ‘worst’ geometric imperfection shape which was adopted to generate these diagrams (see section 6.1.1). The state parameter referred to on the x axis is the vertical deflection of the black bullet depicted in Figure 6.2. . . 128
6.8 All response diagrams at different imperfection amplitudes for the Neum¨unster Abbey courtyard glass roof. On the left the triangular remeshing, on the right our Voronoi (2,2) remeshing. The state parameter referred to on the x axis is the vertical deflection of the black bullet depicted in Figure 6.2. . . 129
7.1 Regular Tilings of the Euclidean Plane. . . 132
7.2 Two sets of alternative coordinates for the stress tensor T in plane stress. . . 134
7.3 Isotropic Triangular Grid, computation of equivalent membrane stiffness. . . 137
7.4 Isotropic Triangular Grid, computation of equivalent bending stiffness. . . 137
7.5 Isotropic Quadrilateral Grid, computation of max. equivalent membrane stiffness.. . . 139
7.6 Isotropic Quadrilateral Grid, computation of min. equivalent membrane stiffness. . . . 140
7.7 Isotropic Quadrilateral Grid, computation of equivalent bending stiffness. . . 142
7.8 Isotropic Hexagonal Grid, computation of equivalent membrane stiffness.. . . 144
7.9 Isotropic Hexagonal Grid, computation of equivalent bending stiffness. . . 147
8.1 Mock-up funicular surface.. . . 154
8.14 (a) Imperfection sensitivity results. According to section 6.1.1, the notations ‘LB’, ‘LS’ and ‘NLS’ within the legend refer to the imperfection shape adopted. (b) GSA vs Straus7, comparison of results. The response diagram for the vertical deflection of point P5 is shown. GSA can only perform incremental proportional loading, nevertheless perfect agreement is experienced with the results yielded by Straus7. On the other hand, taking into account the initial ‘dead load’ results in the load factor being knocked down by around 10%. . . 167
8.15 Numerical and experimental response diagrams for the nodes P1, P2, P3 and P4 dis-played in Figure 8.2 (left), in black−/gray+ and colour respectively. According to sec-tion 6.1.1, the notasec-tions ‘LB’, ‘LS’ and ‘NLS’ within the legend refer to the imperfecsec-tion shape adopted, whereas the number ‘±50’ next to them refer to the maximum signed amplitude (in mm) used for scaling the imperfection field. . . 168
8.16 On the left, a close-up of Figure 8.15. On the right, the most sensible numerical curves together with only the second load test (the most reliable) experimental curve are dis-played for the nodes P1, P2, P3 and P4 (see Figure 8.2 (left)). Although some experi-mental curves are well approximated by certain imperfect numerical curves, no imper-fection shape approximates accurately the behaviour of the whole structure (i.e. of all nodes P1-P4). . . 169
defined in section 5.3.3, respectively. . . 111
6.1 Statistics on datasets. When a reference is given the remeshing comes from that source, otherwise it is a height field isotropic remeshing s(x, y). The acronym Voro (D, A) stands for the statics aware Voronoi remeshing, where D and A are the density and anisotropy user parameters defined in section 5.3.3, respectively. Column BBOX reports the bounding box of the model, so that ε can be computed as defined in section 6.1.1. 121
7.1 Curvature χ and Twist θ along symmetry directions 1, 2, 3 of the triangular grid of Figure 7.4(a), due to a generic state of curvature (χI, χII, χIII). . . 138
7.2 Bending moment MB and Torque MT in the beams of the quadrilateral grid of Figure
7.7. Notice that when two distinct values are listed in the same entry, it means that along that direction one beam is undergoing an internal force and one the other. . . . 143
7.3 Axial force N , shear V and bending moment M within the isotropic hexagonal pattern of Figure 7.8(a), due to the action of a generic state of stress (σI, σII, σIII). . . 145
7.4 Bending moment MB and torque MT within the isotropic hexagonal pattern of Figure
7.9(a), due to the action of a generic state of stress (mI, mII, mIII). . . 148
7.5 Statistics on grid topologies: regular tilings of a planar regual square of 20 m of side with the same overall length. . . 149
7.6 Summary of all the ‘equivalent elastic properties’ (membrane stiffness, bending stiffness, thickness) for the three regular tilings of the plane. . . 150
7.7 Numeric values of all the ‘equivalent elastic properties’ (membrane stiffness, bending stiffness, thickness) for the three regular tilings of the plane. The beams are considered being made of steel and having circular cross section of 50 mm of diameter (EA = 412334 kN , EI = 64.427 kN m2), according to Table 7.5.. . . 150
allora in poi ´e una sfida con te stesso volta a tirare fuori le tue soluzioni migliori, a cercare le collaborazioni giuste, nel tentativo di dare corpo e slancio a quell’idea.
Alla fine, ancora una volta, arriva la scrittura della tesi a chiudere un capitolo della tua vita lungo anni. E evidente allora che questo elaborato ´´ e per te pi´u di un resoconto formale dell’attivit´a svolta: ´e un ‘diario di viaggio’, del quale solo tu conosci la fatica e la cadenza con cui ha preso forma.
In quest’occasione speciale dunque, desidero esprimere la mia riconoscenza verso quelle persone che mi hanno accompagnato in questo cammino.
Innanzitutto grazie al mio tutore Prof. Maurizio Froli, che mi ha introdotto al tema della architectural geometry senza per´o mai forzare la mano, lasciando ampio spazio di manovra alla mia fantasia e tuttavia indirizzandomi al momento opportuno.
Grazie al Prof. Gennaro Amendola, per avermi seguito cos´ı da vicino nei primi sei mesi di stu-dio (quello s´ı davvero ‘matto e disperatissimo’) finanche a sostenermi psicologicamente, nonch´e per il prezioso lavoro di revisione finale.
Un grazie sentito anche ai ricercatori Paolo Cignoni e Nico Pietroni, al Prof. Enrico Puppo ed a tutto il VCLab del Cnr di Pisa, per avermi accolto nel loro laboratorio e per avermi dato la possibilit´a unica di mettere a sistema due discipline, il geometry processing e l’ingegneria strutturale, ancora oggi piuttosto distanti nonostante i tanti sforzi profusi dalla comunit´a sci-entifica.
Grazie ad Alison Martin, artista instancabile e dalla fantasia fervida: le sue creazioni sono state di grande ispirazione sia per me che per il Prof. Froli.
Un sincero ringraziamento va infine al Prof. Chris Williams per avermi gentilmente accordato la straordinaria opportunit´a di raggiungerlo in Inghilterra, cos´ı da collaborare ed al tempo stesso fare nuove conoscenze, migliorare la lingua inglese, venire a contatto con tradizioni e culture nuove: viaggiare insomma.
Ed in quanto ‘diario di viaggio’, ogni riga di questa tesi ´e per me inevitabilmente intrecciata con la mia vita privata. Non posso dunque non menzionare i miei amici d’infanzia: Buzzi, il Defa, Leo, il Biuk e tutti coloro che pur non essendo menzionati sanno di essermi vicini. Gli amici di Pisa: Domenico, Mariano, Orazio, Francesca, Selly, Vincenzo. Quelli di Bath: Dragos, Sotiris, Stella, Daniel, Lizette, Manuel, Gianluca. Le mie gioie e dolori dei giorni passati: Carlotta e Colorado Queen. I miei genitori: grazie babbo e mamma per esserci sempre.
ment of their structural behaviour with respect to state of the art triangular and quadrilateral grid-shells. Later, a theoretical and useful account on the equivalent mechanical properties of triangular, quadrilateral and hexagonal grids is given. Eventually, the digital modeling, build-ing process and load tests on a physical mock-up of a Statics Aware Voronoi Grid-Shell are described.
Chapter 1 introduces the topic of Architectural Geometry. It provides a general description of the subject, it states its main goals and then lists a substantial deal of architectural mas-terpieces that effectively have given rise to the Architectural Geometry itself. Eventually, the key matter of surface discretization is tackled with respect to triangular, quadrilateral and hexagonal topologies.
Chapter2 summarizes the most important concepts of Differential Geometry. Regular curves and surfaces, differentiable mappings, tangent plane, first and second fundamental forms, Gauss Map, principal curvatures and principal directions, Gaussian and mean curvatures, classifica-tion of points on a surface etc... are just some of the introduced concepts. Differential Geometry is a subject of the utmost importance in Architectural Geometry, as the latter always deals with complex (sometimes even free-form) surfaces.
Chapter 3 reviews the main concepts of Computational Geometry, a discipline which turned out to be essential for the digital handling and representation of innovative architectural de-signs. First of all the continuous (NURBS) and piecewise linear (mesh) representations are introduced, followed by the discrete counterparts of some of the geometric notions given in chapter 2. Finally, fundamental concepts such as surface smoothing, parameterization and remeshing are brought up.
Chapter 4 is about the Structural Analysis of shells and grid-shells. At first an outline of the general theory of elastic stability is provided, with a focus on bifurcation, post-buckling analysis and the effect of imperfections. Then the same concepts are reformulated in terms of numerical matrix analysis, that actually represents the only viable tool for the structural analysis of grid-shells. This chapter closes the literature review.
Chapter5presents the framework of the Statics Aware Voronoi Remeshing, the novel remeshing pattern for architecture developed in this dissertation. It starts off by introducing the Voronoi
diagram and its peculiarities, then carries on explaining the basics of the statics awareness algorithm: the linear static analysis of the continuous underlying surface, the local conversion criteria, the remeshing, the regularization, smoothing and symmetrization steps. Eventually, comparisons with quadrilateral state of the art remeshings for architecture are provided. Chapter 6 goes into detail with the structural behaviour of the Statics Aware Voronoi grid-shells. Imperfection sensitivity analyses and stability checks are performed on four contempo-rary architectural models, remeshed with triangular, quadrilateral and (statics aware) Voronoi topologies, respectively. As already stated, the performances of Statics Aware Voronoi free-form Grid-Shells are better than that of state of the art quadrilateral grid-shells.
Chapter7presents a self-contained work about the elastic properties of the three regular tilings of the Euclidean plane. The equivalent membrane stiffness, bending stiffness and thickness of the plate equivalent to each of the three tilings are derived. Then these quantities are col-lected and compared with reference to a likely numerical case study, so that useful results are obtained for the design of grid-shells. In particular, it turns out that the equivalent thickness of polygonal patterns is much higher than that of the triangular pattern, thus justifying their corresponding little sensitivity to imperfections, in total agreement with common knowledge as well as the numerical results of chapter6. Additionally, though isotropic and hence not directly comparable, the equivalent membrane stiffness of the hexagonal pattern is commensurate to that of the quadrilateral pattern.
Chapter 8 describes the realization of a 2.4 m x 2.4 m physical mock-up of a prototype of Statics Aware Voronoi Grid-Shell. At first the digital handling of the design and manufactur-ing processes is illustrated, followed by the practical assemblmanufactur-ing step. Therefore the results of three static incremental load tests are shown, together with some (quite ineffective) attempts to calibrate an accurate numerical model on them.
Eventually, chapter9 provides my personal conclusions about this work and outlines possible future research on the topic.
In particular this process involves two main choices: the selection of an effective paneling tech-nique for the skin, and the optimization of the statics of both the surface itself as well as of the underlying load-bearing structure. Two choices which seem to be totally uncorrelated and that instead are strongly mutually dependent.
1.1
Introduction
Surface Panelization In particular the panelization phase is probably the most critical, as it affects both the aesthetics of the surface, its transparency, its structural effectiveness and last but not least, its fabrication cost.
The panelization phase is mandatory as it represents the only way to build an architectural surface, that is by subdividing it in several pieces. It can also be seen as an “a posteriori optimization”, as it takes the surface as a fixed input. This approach is indeed one of the most fertile in the recent years [GSC+04, LPW+06, ZSW10, EKS+10, PSB+08, CW07, PLW+07, SHWP09,Pot10,FP10].
However, in the early projects (dating back to the ’90s), translational surfaces and scale-trans surfaces have been adopted to approximate complex architectural surfaces (see Figure1.9(a)). Nevertheless the engineers have gradually become aware of the importance of the paneling phase as these methods have proven to be not completely suitable to the design needs. Therefore growing attention has been paid to the development of one’s specific paneling algorithms, often different for each project. Mathematics and geometry have come together in order to create new powerful discretization alghoritms, which at once are able to take into account several types of constraints: geometry, structure, cost, production constraints etc... The tight collaboration, the synergy of several disciplines such as differential geometry, algorithmic mathematics, computer graphics, structural engineering and industry, has eventually given birth to the present field of Architectural Geometry.
This is exactly what arch. eng. Niccol´o Baldassini (director of RFR-Paris) asserts [Bal08]: “(...) a more theoretical approach can expand geometrical knowledge and open a more radical approach to Free-Form design. Research addresses the development of mathematical algorithms: (...) the interest is shifted from the definition to the subdivision of the surfaces. Subdivision is the main point when (...) trying to couple glazing patterns with structural layouts. (...) Free-form design that is sustainable in terms of technologies, costs and aesthetics is in the foreseeable future. Mathematics, geometry, technology and production are all converging together.”
Surface Optimization Free-form surfaces are first of all three-dimensional surfaces, shells in the technical jargon. Shells have additional stiffness thanks to their shape, they are intrin-sically effective structures, but on the other hand they are very sensitive to imperfections both in the design and in the fabrication steps.
Since the very beginning of the XX century and even before, engineers have tried to enhance shells’ properties in many different ways: for example by building physical models or through complex analytical calculations. Just think of names such as Nervi, Morandi, Musmeci, Tor-roja, Gaud´ı, Candela, Dieste and their amazing creations (see Figures 1.1, 1.2, 1.3, 1.4, 1.5,
1.6,1.7).
All these attempts to improve the properties of the surface can be addressed as an “a priori optimization”, as for them the surface is not an input but an output. This approach is in net contrast with that of section1.1, but they share the common end of trying to reach an optimal solution. The former often in terms of the statics of the surface, the latter in terms of its paneling.
Usually an optimal project requires that an “a priori optimization” is followed by an “a posteri-ori optimization”, often in an iterative way, in order to obtain an optimal result. Nevertheless, recently some attempts have been done in order to merge this two steps into one single proce-dure [TSG+14].
(a) Hangar in Orvieto (IT), 44.8 m by 115 m, con-crete, completed in 1935 (now destroyed).
(b) Hangar in Orbetello (IT), steel, completed in 1941 (now destroyed).
(c) Sport hall, Rome (IT), completed in 1960. (d) Hall Paolo V I, Rome (IT), completed in 1971. Figure 1.1: Some of Pierluigi Nervi masterpieces.
(a) Bridge, Catanzaro (IT), max span 231 m, completed in 1962.
(b) Bridge on Maracaibo Lake, Venezuela, max span 235 m, started in 1957.
Figure 1.2: Some of Riccardo Morandi masterpieces.
(a) Bridge, Potenza (IT), completed in 1976. (b) Ceiling of the “Tent Church”, Vi-cenza (IT), completed in 1962.
Figure 1.3: Some of Sergio Musmeci masterpieces.
(a) Market shell roof, Algeciras (ES), completed in 1934.
(b) Racecourse roof, “La Zarzuela” Madrid (ES), completed in 1935.
(a) Crypt, Park Guell, Barcelona (ES), completed in 1915.
(b) Mil´a house inverted catenary, Barcelona (ES), completed in 1912.
Figure 1.5: Some of Antoni Gaud´ı masterpieces.
(a) Caf Los Manantiales, Xochimilco (Mexico), completed in 1958.
(b) Pavilion of cosmic rays, Mexico, completed in 1951.
(c) Bacardi bottling plant, Cuautitlan (Mexico), completed in 1960.
(d) F´abrica Celestino Fern´andez, Colonia Vallejo (Mexico), completed in 1955.
Figure 1.6: Some of F´elix Candela masterpieces.
(a) Free standing barrel vault ‘The seagull’, Salto (Uruguay).
(b) Church of Christ, Atlantida (Uruguay), completed in 1952.
project by arch. Norman Foster, engineering by Buro Happold and Prof. Chris Williams, con-struction by Waagner Biro. Then some other intriguing shapes with triangular connectivity in Figure1.12, up to the amazing and challenging skyscrapers of Figure 1.13. In Figure 1.14
instead are depicted two among arch. Gehry’s masterpieces: the former with a metal skin, the latter with an astonishing glass skin, the clear result of fifteen years of research and innova-tion in the field of Architectural Geometry. Then Figure 1.15 shows two delightful projects engineered by RFR-Paris, the latter (the Strasbourg train station) been built with cold-bent glass: another achievement of applied research in architecture. Figures 1.16(b) and 1.19 in-stead depict two among the biggest steel-glass grid-shells ever built: the former in Abu Dhabi, the latter in Singapore. Finally five examples, actually the only I know so far, of hexagonal and polygonal grid-shells. The Eden Project (Figure 1.20), a greenhouse in Cornwall, is the oldest as well as the most famous. The two examples of Figure1.21instead are both from the japanese architect Shigeru Ban, obtained as a height of field of a semi-regular tiling. Figure
1.22shows the roof of the Shidare observatory in Kobe, an astonishing geometry based on a Voronoi-like topology (see section5.1) and built through the reciprocal frame concept. Finally, Figure 1.23 shows a great achievement obtained at the University of Minnesota in 2012: a thin-shell corrugated vault tessellated by an anisotropic hex-dominant pattern.
Eventually, in Figure 1.24 some astonishing examples of contemporary ‘free-form reinforced concrete shells’ by japanese engineer Mutsuro Sasaki are shown. Though a bit off topic, I crave to acknowledge his work as I was stunned by his presentation held at the conference AAG 2012 in Paris. Additionally, it is worth mentioning his works because of their uniqueness and their novel ‘balance between engineering knowledge and visual expression’ [ABVW14, p. 261], the rationality of the shape being determined by the minimization of global strain energy, just as in the present work (see chapter5).
As a last remark, for a thorough review about the history and development of grid-shells and the use of glass in buildings, the reader is referred to the useful essays of Prof. Maurizio Froli [BFL06,GF09,FBC09,BFL06].
(a) Hamburg History Museum Court Roof. (b) Hyppo House Berlin Zoo.
Figure 1.9: (a) Hamburg History Museum Court Roof, Hamburg, completed in 1989. (b) Hyppo House, Berlin Zoo, completed in 1996.
(a) DG Bank Court Roof. (b) Maritime Museum Osaka.
Figure 1.10: (a) DG Bank Court Roof, Berlin, completed in 1998. (b) Maritime Museum, Osaka, completed in 2000.
(a) British Museum Great Court Roof. (b) British Museum Great Court Roof. Figure 1.11: British Museum Great Court Roof, London, completed in 2000.
(a) BMW Welt. (b) Blob.
Figure 1.12: (a) BMW Welt, Monaco, completed in 2007. (b) Blob, Eindhoven, completed in 2010.
(a) The Gherkin. (b) Aldar Headquarters. (c) Capital Gate.
Figure 1.13: (a) The Gherkin, London, completed in 2003. (b) Aldar Headquarters, Abu Dhabi, completed in 2010. (c) Capital Gate, Abu Dhabi, completed in 2011.
(a) Guggenheim Museum. (b) Foundation Louis Vouitton.
Figure 1.14: (a) Guggenheim Museum, Bilbao, completed in 1997. (b) Foundation Louis Vouitton, Paris, completed in 2012.
(a) Ne¨umunster Abbey Court Roof. (b) Strasbourg Train Station Glass Facade.
Figure 1.15: (a) Ne¨umunster Abbey Court Roof, Luxembourg, completed in 2006. (b) Stras-bourg Train Station Glass Facade, StrasStras-bourg, completed in 2008.
(a) Cabot Circus Grid-Shell. (b) Yas Hotel.
Figure 1.16: (a) Cabot Circus Grid-Shell, Bristol, completed in 2008. (b) Yas Hotel Grid-Shell, Abu Dhabi, completed in 2009.
(a) Vela Fiera Milano-Rho. (b) Islamic Art Pavilion.
Figure 1.17: (a) Vela Fiera Milano-Rho, Milano, completed in 2005. (b) Islamic Art Pavilion, Louvre Paris, completed in 2012.
(a) Smithsonian Institution Court Roof. (b) Zlote Tarasy Shopping Centre.
Figure 1.18: (a) Smithsonian Institution Court Roof, Washington DC, completed in 2007. (b) Zlote Tarasy Shopping Centre, Warsaw, completed in 2007.
Figure 1.19: Gardens by the Bay, Singapore, completed in 2014.
(a) Eden Project. (b) Kreod Pavilion.
Figure 1.20: (a) Eden Project, Cornwall, completed in 2000. (b) Kreod Pavilion, London (temporary installation), completed around 2011.
(a) Nine Bridges Golf House. (b) Centre Pompidou-Metz.
Figure 1.21: (a) Nine Bridges Golf House, Seoul, completed in 2010. (b) Centre Pompidou, Metz, completed in 2010.
(a) Rokko Shidare Observatory. (b) Rokko Shidare Observatory.
Figure 1.22: (a) Rokko Shidare Observatory, Kobe, completed in 2013.
(a) Catalyst Hex-Shell. (b) Catalyst Hex-Shell.
Figure 1.23: (a) Catalyst anisotropic hexagonal corrugated thin-shell, Minneapolis, mock-up completed in 2012.
(a) Kakamigahara Crematorium.
(b) Rolex Learning Centre.
(c) Teshima Art Museum.
Figure 1.24: Mutsuro Sasaki masterpieces. (a) Kakamigahara Crematorium, Gifu 2006. (b) Rolex Learning Centre, Lausanne, 2010. (c) Teshima Art Museum, Kagawa, 2010.
1.2
Surface Discretization
The terms discretization and panelization are synonyms. As already said in section 1.1, cur-rently the panelization issue is tackled through geometric algorithms. But before choosing or developing one’s specific algorithm, it is mandatory to decide on a certain topology. Topology is probably the most important variable when dealing with grid-shells, as it will be shown in chapters5 and7.
Currently the most commonly adopted topologies are the triangular and the quadrilateral, and seldom also the hexagonal one. All of them show some pros as well as some cons, as briefly summarized in Table1.1.
Surface Faces Planariz. Valence of Torsion Overall Sensitivity to Approx. Complexity Reg. Nodes of Nodes Stiffness Imperfections
Tri Optimal Intrinsically Flat 6 Yes High High
Quad Good Quite Easy 4 No Low Low
Hex Quite Good Not trivial 3 No Low Low
Table 1.1: Comparing grid-shells’ topologies main properties.
1.2.1
Triangular Topology
Table 1.1 clearly shows why the triangular topology was the first in time to be employed in architectural geometry. Just think of the geodesic domes of Buckminster Fuller (see Figure
1.25(a)):
1. the approximation of the surface is excellent since each triangle can be arbitrarily locally refined into smaller triangles, without creating any change nor in the topology neither in the static scheme (see Figure1.25(b));
2. even if the faces are very different in size and aspect ratio, the panels will always be flat; 3. high overall stiffness of the structure;
4. broad availability of algorithms for triangular subdivision, thanks both to the computer games industry and to the C.A.E./C.A.M. softwares research.
(a) Geodesic Dome. (b) Local refinement of triangular grid.
Figure 1.25: (a) Buckminster Fuller geodesic dome in Montreal, completed in 1967. (b) Local refinement of a triangular grid topology.
(a) Geometric Torsion in a Triangular Grid.
(b) Geometric Torsion in a Node of the project of Figure1.17(a).
Figure 1.26: (a) Geometric Torsion in a Triangular Grid. (b) Geometric Torsion in a Node of the project “La Vela - Milano Fiere” (see Figure1.17(a)), Milano (IT).
Excellent examples of triangular grid-shells are shown in Figures 1.10, 1.11and 1.12, respec-tively.
Nevertheless triangular meshes exhibit also several cons, including high valence as well as ge-ometric torsion of the nodes (see Figure 1.26). Essentially for these reasons nowadays the triangular topology is used less and less, most of the times being replaced by the quadrilateral one. Therefore, presently designers resort to the triangular topology only if there are special needs for stiffness and strength of the structure.
1.2.2
Quadrilateral Topology
With respect to triangular meshes, quadrilateral meshes exhibit two remarkable disadvantages: a lowest stiffness and generally speaking non planar panels.
Whereas it is not possible to obtain the same stiffness of an equivalent triangular grid, the flexibility of a quadrilateral grid can be lowered by properly orienting the members with respect to the load patterns and the external restraints (see Figure1.27and chapter7).
For what concerns the Gaussian curvature (see chapter2) of the panels instead, flat or single curvature panels can be achieved by applying a suitable quadrilateral subdivision algorithm. Incidentally, flat panels are of course cheap to produce, but also single curvature panels can be
(a) Squares. (b) 45◦Squares.
obtained at little cost through the cold bending technique [DT07,ES07,VG04,BIVIC07], the same which was adopted in the project of the Strasbourg train station (see Figure1.15(b)). Currently several quadrilateral subdivision algorithms are available: some of the most renown and effective will be discussed in the following.
Scale-Trans Meshes At first (i.e. until the 90’s), the quadrilateral subdivision of surfaces was tackled through purely geometric methods. The continuous surface was approximated by means of quadrilateral facets, obtained via translational and rotational motions, and combi-nations of the two as well. This approach was particularly dear to Sclaich Bergermann und Partner, who adopted it in several of their projects [GSC+04,Sch05] (see Figure1.9).
Figure1.28shows the main steps of the procedure:
for each “row” composing the mesh, a set of parallel vectors with possibily different lengths is choosen. Hence the planarity of the faces is guaranteed indeed (Figure1.28(a)); by adding “rows” along a certain curve called “directrix”, the mesh is built (Figure
1.28(b));
ruled surfaces can also be obtained as a particular case of scale-trans surfaces (Figure
1.28(c));
complete “scale-trans” meshes can be created by combining the translational technique with expansions of the cross sections and a three-dimensional directrix curve.
(a) Translational mesh - basics. (b) Translational mesh. (c) Ruled surface obtained as a translational surface.
(d) Longitudinal sections and concentric expansion. (e) Complete “scale-trans” surface. Figure 1.28: Scale-Trans Meshes [GSC+04], [Sch05].
PQ perturbation algorithm In 2006 a useful and powerful quadrilateral subdivision algo-rithm appeared: the ‘planar quad (PQ) perturbation algoalgo-rithm’ [LPW+06]. This algorithm interlaces two main phases: the first step is a classical quadrilateral subdivision of surfaces per-formed through standard algorithms like Catmull-Clark [CC78] or Doo-Sabin [DS78], whereas the second step forces a perturbation of the vertices’ position aimed to make faces planar. By iteratively applying steps 1 and 2 the algorithm produces fine planar quad-dominant meshes from coarse non planar quad meshes. Figure1.29(a) shows the result of the application of PQ algorithm on a coarse quadrilateral starting mesh.
property endows the conical meshes with an orthogonal supporting structure. This means that the symmetry plane of each beam composing the structure corresponds with the plane passing through the edges in common between two faces and their offsets, respectively.
(a) PQmeshes hierarchy, obtained by repeatedly ap-plying the PQ perturbation algorithm.
(b) Faces configuration at a vertex of a conical mesh.
Figure 1.29: PQ perturbation algorithm and conical meshes [LPW+06].
TCD Fields (transverse conjugate directions fields) Networks of conjugate curves on a surface Φ are two families A, B of curves which completely cover the surface Φ. They are such that for each point P of Φ there passes only a curve of family A and one of B. Additionally, these two curves are such that their tangents are conjugated at P . The conjugacy relationship between two directions on a surface is explained analytically at point30of chapter2, although a geometric construction is also possible. Figure1.30shows this construction for all possible kinds of points on a surface.
Each surface has more than a couple of conjugate curves: for example the network of principal curvature lines is always conjugate, and in addition the curves intersect always at 90°. An important property of conjugate curves is that the envelope of the tangent planes to a conjugate curve belonging to the family A is a developable surface, whose rulings are tangent to the corresponding conjugate curve of the family B. It is evident therefore that if the polylines of a PQmesh are drawn following a network of conjugate curves, then it is much easier to achieve the planarity of the faces.
This is why [ZSW10] focuses the attention on the research of networks of conjugate curves on a surface, as this knowledge endows the grid-shell designer with more freedom and at the same time ensures PQmeshes of better quality.
D-Strips (developable strips) Developable strips [PSB+08] represent the refining of a PQmesh strip along the longitudinal direction (see Figure1.31). D-strips allow to discretize the reference surface with single curvature panels, rather than with flat panels. Therefore they have a better approximation power with respect to planar quad meshes, but at the same time they are developable surfaces and hence are not expensive to produce.
Figure 1.30: Geometrical construction of the conjugacy relationship at a point P of the surface Φ, respectively hyperbolic, elliptic and parabolic [ZSW10].
(a) Developable strips obtained by refining rows of PQmeshes.
(b) Semi-discrete meshes as limit case of discrete meshes.
Figure 1.31: Developable strips [PSB+08].
also the use of cold-bent glass [DT07,ES07,VG04,BIVIC07]. Nevertheless, there are already examples where cold-bent developable stripes have been aligned to the lines of maximum curvature of the surface (see Figure1.15(b)).
Paneling Algorithm Further studies have also addressed the cost and the fabrication issues of paneling large free-form surfaces with quadrilateral facets. This is a very interesting topic, as in large scale free-form projects it is generally not possible to achieve planarity or single curvature everywhere on the surface. On the other hand the overall cost of the skin is highly determined by the curvature of its panels, the double curved ones being definitely the most expensive. From this framework there comes out an algorithm which takes in input the number of moulds available for the panels (e.g. flat, several kinds of single and double curvature etc..) and the surface approximation tolerances, and then gives in output the optimal quad-dominant mesh which minimizes the overall cost while maximizing the approximation of the input surface (see Figure1.32).
1.2.3
Ruled Surfaces
Ruled Surfaces are a specific class of surfaces with single or double curvature, such that their Gaussian curvature (see section 28) is always less than or equal to 0 (K ≤ 0). Examples of ruled surfaces are the plane, the cylinder, the cone, the hyperbolic paraboloid, the helicoid, the tangent developable surface of a spatial curve etc... They are said to be ‘ruled’ because they are actually composed of straight lines.
Among all the ruled surfaces, those with zero Gaussian curvature (K = 0, i.e. the cylinder and the cone) are also developable, which means that they can be flattened onto a plane without distortion [Wik14c]. This property is of utmost importance when it comes to paneling a sur-face, as its portions with zero or slightly negative Gaussian curvature can be well approximated by developable surfaces.
devel-(a) Global cost comparisons by using the Paneling algorithm. (b) Terminology and variables adopted in the algorithm. Figure 1.32: Paneling algorithm [EKS+10].
opable surfaces. Hence the panels which cover these parts can be easily produced from planar sheets of material, and then bent into shape on the stocks (see Figure1.33).
The pipeline of the approximating process can be summarized as in the following [FP10, Pot10]):
1. compute the Gaussian curvature everywhere on the surface and skim those unsuitable regions where K > 0;
2. compute the asymptotic directions (directions along which the normal curvature is zero, see point 27 of chapter 2) in several points. Those are the directions along which the rulings of the attempt ruled surface must be aligned;
3. choose the most suitable ruled surface for each patch, by minimizing the total squared distance between the attempt and the reference surfaces.
Figure 1.33: Design for the Cagliari contemporary art museum. Paneling of certain portions of the skin through ruled surfaces [FP10].
1.2.4
Hexagonal Topology
Just like quadrilateral meshes, hexagonal meshes exhibit a low overall stiffness with respect to an equivalent triangular grid (see chapter 7) and non planar panels in general. On the other hand they are aesthetically pleasing, captivating, most of the times they even resemble organic
forms, and additionally they have a very low valence of the nodes which makes their production much easier.
Stiffness can be improved in several ways, for example by introducing stiffening devices like bracing cables ([SS94, SS96, SS02]) or by shaping the cells according to the statics of the surface (see chapter5). Hence stiffness shouldn’t be regarded as a main issue for the hexagonal topology.
Instead, generally speaking, it is not straightforward to enforce the planarity of the panels. Indeed it is more difficult than in the quadrilateral case, as most of the available planarization techniques which work well for the quadrilateral topology [BDS+12], lead to some overlapping vertices when dealing with hexagonal faces.
Although there are no hexagonal remeshing algorithms capable of enforcing planarity yet, some hexagonal or hex-dominant remeshing algorithms are already at hand and will be discussed in the following.
Honeycomb subdivision The honeycomb subdivision algorithm [AS03] is a remeshing op-erator which translates a triangular mesh into an hex-dominant one. The basic steps are shown in Figure1.34(a):
the starting point is a (even unstructured) triangular mesh, therefore coarse quad meshes like those in Figure1.34(b) must be triangulated before starting off with the process; two vertices are laid down per each edge, one on each side of the edge;
vertices within each triangular face are connected by edges. In regular (triangular) faces this leads to new triangular faces;
each couple of vertices corresponding to the same edge is connected. Now there are no unconnected vertices on the mesh;
as a refinement step, the small new triangles are collapsed into a point. The connectivity of the mesh gets simplified and the average size of the cells becomes more uniform. Some results are displayed in Figure1.34(b): it is evident that the cells are all much different one from another, but the result is highly organic and aesthetically pleasing.
Heuristic Method ‘Heuristics’ are: “(...) experience-based techniques for problem solving, learning, and discovery that find a solution which is not guaranteed to be optimal, but good enough for a given set of goals. Where the exhaustive search is impractical, heuristic methods are used to speed up the process of finding a satisfactory solution via mental shortcuts to ease the cognitive load of making a decision. Examples of this method include using a rule of thumb, an educated guess, an intuitive judgment, stereotyping, or common sense” [Wik14f].
In [CW07] an heuristic method for generating planar hex-dominant and polygonal remeshings in general is presented. Figure1.35(a) shows the main steps of the procedure:
starting off from a triangulated mesh, an user defined number of seeds is chosen over it; according to an user specified metric, triangles are clustered around the prescribed seeds
into regions;
planes interpolating the regions are drawn and their mutual intersections give birth to the edges and to the vertices of the polygonal mesh, respectively.
As shown in Figure 1.35(b), very different results can be obtained according to the adopted metric. In case, also organic-like patterns can be achieved.
(a) Steps of the algorithm. (b) Examples obtained with the Honeycomb Subdivi-sion.
Figure 1.34: Honeycomb subdivision algorithm [AS03].
(a) Steps of the algorithm.
(b) Organic-like patterns obtained with particular user-specified metrics.
(c) Natural animal skin patterns (elephant and snake from left to right, respectively).
Figure 1.35: Heuristic planar hex-dominant remeshing algorithm [CW07].
TPI algorithm (tangent plane intersection algorithm) The tangent plane intersection algorithm [Tro08] is actually another remeshing operator which converts a triangular mesh into a hex-dominant one. The starting point of the process is still a triangular remeshing of the
reference surface, the result of it being highly dependent on the triangulation adopted. For each vertex of the triangulation the tangent plane to the surface is drawn, hence from each triangle a point is obtained as the intersection point of its three corresponding tangent planes. This vertex belongs to the final hex-dominant remeshing. By repeating the process for each triangle of the starting mesh, the complete hex-dominant remeshing can be worked out. Figure 1.36(a) shows the influence of the Gaussian curvature K over the geometry of the remeshed panels: it is evident that convex polygons appear on positive curvature zones, whereas bow-tie shaped faces correspond to places with negative Gaussian curvature. Figure 1.36(b) instead displays the result of the algorithm over a non-optimal initial triangulation.
(a) Gaussian curvature implications on the panels’ shape. (b) Result of the TPI algorithm. Figure 1.36: TPI algorithm [Tro08].
EO meshes (edge offset meshes) As already explained about circular and conical meshes, an ‘offset mesh’ is a mesh located at a certain constant distance from its ‘mother’, with exactly its same connectivity. As a mesh is not a continuous surface, more than a distance can be defined and therefore different offset meshes exist: vertex offset, face offset and edge offset meshes, respectively.
The paper [PLW+07] is all focused on edge offset meshes, as they play a fundamental role in building an effective load-bearing structure. In fact only edge offset meshes allow to build a frame with constant height beams (see Figure 1.37(b)). Moreover, as their symmetry planes intersect at nodes in a common axis, edge offset meshes also provide nodes without geometric torsion.
Nevertheless, the paper shows that the edge offset meshes are a very restrictive category, as they are mainly obtained by applying specific mathematical transforms to certain input meshes (see Figure1.37(c)). Then the edge offset meshes should be used in the design process, rather than in the following rationalization phase of a free-form reference surface.
CP meshes (circle packing meshes) Just as the honeycomb and the TPI algorithms, also the circle packing meshes [SHWP09] exploit the duality that binds the triangular and hexagonal tilings. But what distinguishes the CP meshes from the two aforementioned algorithms is the way it computes the starting triangulation.
They enforce all the triangles to have a circle inscribed, such that adjacent triangles have their circles touching on the shared edge (see Figure1.38(a)). The circle packing condition is only obtainable on cylinders and toruses, whereas on other surfaces only approximations can be achieved.
It is also worth noticing that convex hexagons are obtained also in zones with negative Gaussian curvature (this is not the case for the TPI algorithm for example). CP meshes deliver also (geometric) torsion-free nodes, which is a really useful property. Another interesting fact is that circle packing meshes yield several engaging hybrid meshes, such as the tri-hex shown in Figure 1.38(c), but also others like quad-octa and more, by simply varying the starting triangular CP mesh.
(c) EO mesh generated applying a Laguerre transform to the hexagonal meshing of a Koebe polyhedron. Figure 1.37: Edge offset meshes [PLW+07].
(a) Two contiguous triangles in a CP mesh: the circles touch each other on the shared edge.
(b) Overall view of a CP mesh (foreground) and its dual (back-ground).
(c) Tri-Hex hybrid mesh, resulting from a CP mesh. Figure 1.38: CP meshes [SHWP09].
Free-form honeycomb structures Another purely hexagonal remeshing algorithm is pre-sented in [JWWP14]. Here a procedure for deriving honeycomb structures with edges which meet at 120°(see Figure1.39(a)) is thoroughly described.
degrees of freedom. This means that it allows the user for the implementation and the op-timization of several constraints at the same time. The only hard constraint they point out is the orthogonality of the supporting structure with respect to the reference surface, which cannot be reached everywhere unless the surface is developable [Wik14c].
Apart from this, torsion-free support structures can be achieved, with constant height beams and high standardization of both beams’ length and nodes’ shape. Additionally, several new patterns can be obtained from a honeycomb structure, by simply applying ‘graphical stencils’ (see Figures1.39(b) and1.39(c)). The hexagonal faces are still not planar, even though sev-eral ways have been set up in order to split them into planar quads (see Figures 1.39(d) and
1.39(e)).
(a) Overall view of a free-form honeycomb structure.
(b) “Graphical stencils” for obtaining new derived subdivision patterns.
(c) “Graphical stencils” for obtaining new derived sub-division patterns.
(d) Splitting a hex non-planar face into two quad planar facets.
(e) Splitting a hex non-planar face into two quad planar facets.
dissertation more self-contained, keeping at hand the essential concepts which result needful in the following chapters. The whole exposition keeps a low profile, it is not always strictly rigorous and neither proofs nor practical examples are given: as previously stated, the aim is just that of acquainting a possible beginner reader with the most important concepts about the geometry of curves and surfaces.
My learning about this topic was largely supervised by Prof. Gennaro Amendola, and I would like to thank him for his most useful guidance, as well as for the helpfulness and good will he has always shown to me.
As this is not a thorough work on differential geometry but rather a recap, subsections are avoided and each concept is simply outlined in separate points with a consecutive numbering. Some of these points deal with fundamental concepts, therefore they are labeled with a F symbol so that the reader can quickly locate them amongst others. The rest of them usually handles the mathematics necessary in order to state subsequent concepts, or it is just an in-sight; thus roughly speaking they can be omitted at a first reading.
Regular curves are tackled in section2.2, whereas some aspects of the instrinsic geometry of regular surfaces are dealt with in section2.3. Section2.4instead is all devoted to the topic of differential operators defined on surfaces, such as the Laplace-Beltrami operator which will be used in chapter3.
Most part of the material reported in this chapter, including Figures, is from the book ‘Dif-ferential Geometry of Curves and Surfaces’ [dC76]. Nevertheless some additional material has also been taken from [Pis65,DJK+10,dC92,HLS07,Sch14,Can14,Wik14i,Wik14g,Wik14e], therefore in the following proper references are provided just beside each point title, so that the reader can easily have a look to those textbooks for further details.
2.2
Regular Curves
1. Maps [dC76] A map is a vector valued function, that is a function whose image is three-dimensional (or, in general, n-three-dimensional), as compared to a scalar function whose range is one-dimensional. The input of a map could be respectively a scalar or a vector, as shown in
the following examples:
x(t) = (x(t), y(t), z(t))
x(u, v) = x(u, v), y(u, v), z(u, v)
A map x is continuous if all its component functions are continuous. A map x is differentiable if all its component functions are differentiable.
2. Regular Curves A parametrized differentiable curve is a single parameter differen-tiable map α : I → R3 (see point1
) of an open interval I = (a, b) of the real line R into R3.
3. Tangent vector of a Regular Curve [dC76, p. 2] F Let α(t) with t ∈ It⊂ R and
α(s) with s ∈ Is ⊂ R two parametrization, respectively by a generic parameter t and by arc
length s, of the same differentiable regular curve C ⊂ R3. Then both the vectors:
α0(t) = x0(t), y0(t), z0(t), |α0(t)| =px0(t)2+ y0(t)2+ z0(t)2
α0(s) = x0(s), y0(s), z0(s), |α0(s)| = 1
are its tangent vector (or velocity vector ). It is evident that, in general, only the arc length parametrization α(s) has a unitary tangent vector.
4. Curvature of Regular Curves F [dC76, p. 16-17] Let α(s) be the parametrization by arc length of the regular differentiable curve C ⊂ R3, then the vector α00(s) = k(s)n(s) is
the derivative of the tangent vector α0(s) (see point3) at s. The scalar |α00(s)| = k(s) is called the curvature of α at s, whereas the unit vector n(s) is called the principal normal of C at s. Additionally, the inverse of the curvature R = k1 is called the radius of curvature.
It is worth noticing that, since the derivative of a vector with constant module is orthogonal to itself (2hn0(s), n(s)i = dsdhn(s), n(s)i = d
dt(1) = 0), the curvature vector α
00(s) is always
orthogonal to the velocity vector α0(s).
If the curve C is not parametrized by arc length, but rather by a generic parameter t, the value of the curvature obviously keeps the same. Nevertheless, its analytic expression gets a bit more complex.
5. Curvature of Planar Regular Curves in form of Graphs F [Pis65, p. 219-220] As a particular case, let C ⊂ R2 be a regular planar curve, which is given in form of a graph y = f (x) rather than by means of a map α(t). In this case the expression of the curvature k(s) (see point4) can be carried out in analytic form (see Figure2.1):
k(s) = lim∆s→0 ∆ϕ ∆s = dϕ ds ϕ = arctandy dx, dϕ dx = d2y dx2 1 +dy dx 2 k(s) =dϕ ds = dϕ dx ds dx = d2y dx2 1 +dy dx 2 r 1 +dy dx 2 = d2y dx2 h 1 +dy dx 2i3/2
6. Torsion of Regular Curves F [dC76, p. 18] Let α(s) be the parametrization by arc length of the regular differentiable curve C ⊂ R3, with t(s) and n(s) respectively its tangent
