, , vol.180, p.183

,

187 7.2.1 Types of two-colouring ,

197 7.3.1 Third-colour deletion ,

,

, , p.226

, , p.269

271 10.2.3 Mapping and clustering for aided exploration ,

,

, ATATA INSERT(T,4): ATPTTA INSERT(T,2): ATTPTA INSERT(D,3): ATPDTA SUBSTITUTE, vol.3

92 4.13 Skeleton-based decomposition of a surface in a coarse quad mesh to yield a pattern aligned with the boundary and point features off the boundary, vol.14, 2015. ,

, 96 4.16 Form diagram ?, force diagram ? * and form-found network G for the Rhön-Klinikum Campus cable net using Thrust Network Analysis [Block and Ochsendorf, p.97, 2000.

, 18 Skeleton-based decomposition of a surface in a coarse quad mesh to yield a pattern aligned with curve features disconnected to the boundary, vol.98, p.99

, Additional steps for skeleton-based generation of coarse quad meshes including curve features. Curve features are unwelded to avoid the crossing skeleton branches in dashed pink curves. Pentagonal or higher-valency faces, in blue, resulting from the unwelding become quad faces by propagating the seams of the discrepancies on the curve features

, Exploring the topology of a quad-mesh pattern using skeletonbased decomposition with different curve features, p.100

, Each cross field direction corresponds to one of the two groups of elements, in red or blue. The patterns stem from the principal stress directions for different loading conditions, Integration of cross fields yields two-coloured patterns by definition

, The three types of two-colourability for quad-mesh patterns, vol.188

, Singular vertices V become singular faces F * in the dual mesh, which is therefore not a quad mesh but a quaddominant mesh, Dual relation in a quad mesh M and its dual M * between topologically continuous polyedges P and strips S, in blue and red

, Quad mesh with two-coloured polyedges but without twocoloured vertices or faces due to the odd number of elements along the closed strips

, Quad mesh with two-coloured vertices but without two-coloured faces or polyedges due to the odd number of elements around the singularity

, Quad mesh without two-coloured vertices, faces or polyedges due to the odd number of elements along the closed strips and around the singularity

, The vertex and face elements along an open strip can always be two-coloured

, The vertex and face elements along a closed strip can be twocoloured if crossed by an even number of elements, p.192

, 192 7.12 The even numbers of strips around the non-boundary vertex in pink and along the boundary guarantee the fulfilment of the singularity requirement for two-colouring, p.193

, Non-two-colourable coarse quad meshes because of odd numbers of strips

, Iterative two-colouring of the vertices of a graph, p.195

with 2N vertices connected to N vertices, resulting in a quadratic time complexity O(n 2 ) for the two-colouring algorithm, vol.197, p.197 ,

, Quad mesh with two-coloured polyedges after deletion of the third-colour strips, evidenced by new two-coloured strip graph of its coarse quad mesh

20 This graph represents the search organisation through the combinations of three deletion rules 1, 2 and 3. The search algorithm applies only deletion rules, defining an orientation the upstream combination (-) to the downstream combination (1,2,3). For instance, (1,2) is in the downstream direction of (1) but (2,3) is not in the direction of (1), The combinatorial richness in colouring a graph using more than two colours provides multiple combinations to delete third-colour strips ,

, Deleting a second strip of a two-coloured quad mesh is redundant because it yields another two-coloured quad mesh, but at a farther distance and in the same direction, p.202

, Deleting two strips separately may not yield a two-coloured quad mesh, but can yield a two-coloured quad mesh at a farther distance in a new direction

, Principle of the search algorithm for two-coloured quad meshes. The search yields only the closest two-coloured quad meshes in independent design directions. The two-coloured quad meshes are highlighted in pink, and the discarded quad meshes downstream are marked by dashes

204 7.26 A coarse quad-mesh that does not comply with the twocolouring requirements. The five-valent singularities does not allow to fulfil the singularity requirement, as evidenced by the three-coloured strip graph and the index-based characterisation, p.207 ,

, 28 Two-colour projection applied to a seven-strip quad mesh with a hexagonal shape, coarse quad meshes resulting from two-coloured topology finding and their corresponding checkerboard tilings with different singularities

Two-colour projection applied to a nine-strip quad mesh with a rectangular shape with one opening, p.211 ,

Two-colour projection applied to a 14-strip quad mesh with a rectangular shape with two openings, p.212 ,

, Folded patterns increase the bending stiffness along the folds and decrease the membrane stiffness across the folds, p.214

, Principle of the folding pattern of the CNIT. A six-valent singularity in pink provides a two-coloured pattern for stiffening the directions that carry the loads along the free edges directly to the supports in green. The stiff primary direction is in black and the secondary direction in grey, p.215

, Topology from skeleton-based decomposition of the surface with labelled strip. The resulting pattern can not be twocoloured due to the four five-valent non-boundary singularities, p.217

218 7.36 The five smooth parameterisation quad meshes patterns with two-coloured polyedges ,

, The ten folded shells resulting from the five pattern topologies times the two folded directions, marked as X and Y in the X-Y labels, respectively

, Level one of subdivision, highlighted in red, provides sufficient precision for this analysis without resorting to a too large number of faces, Study of the convergence of the strain energy for the smooth design of topology 1, p.221

, Influence on the strain energy of introducing folds with different directions and amplitudes. Results above twice the initial strain energy are not shown, as evidenced by the dashed curves, vol.222

The covariance in kN 2 .m 2 of each strip with the structural performance when stiffened in the folded designs highlights their positive or negative influence ,

226 7.43 2k-valent singularities are adjacent to odd number of strips per colour, which does not provide the symmetry for matching directions, represented by the red arrows, between the strips of the same colour, opposed to 4k-valent singularities, which are adjacent to even number of strips per colour ,

, The quad-mesh grammar allows deleting and adding the strips in blue. However, the grammar can not modify a strip, despite the similarity between the old and new strips, p.235

, Halfedge mesh data structure with edges producing oriented halfedges, in blue, pointing to the adjacent faces. The elements around the faces and the vertices are ordered and sorted in opposite directions, in orange, p.239

, Moving a marker forward is ambiguous: for a halfedge directed towards a n-valent vertex, as n-1 other halfedges may be chosen

, 240 8.5 Extension of the two movement operations with a parameter for multiple application of the operation, p.241

, of the marker belongs to. The marker is beforehand positioned on the polyedge resulting from the deleted strip, The delete operation deletes the strip to which the halfedge

, Therefore, the marker is moved using the pivot operation with parameter -1, to bring the marker on the new polyedge, When applying the delete operation, the pivot operation on the marker, in blue, may not yield a halfedge on the new polyedge, in red

, Turning it on initiates the collection of a polyedge in red as the successive positions of the marker in blue. Turning it off adds a strip along the polyedge and empties the collected polyedge

, Extension of the add operation with parameters to add closed strips and strips with poles

244 9.2 From the data structures of meshes to coarse pseudo-quad meshes for topology finding of quad-mesh singularities. Data is added to integrate the strip structure in quad meshes, density parameters in coarse quad meshes and pole singularities in pseudo-quad meshes, The strips can be added progressively during the collection of the polyedge by adding a face of the strip after each turn movement ,

, Normalised covariance between topology and performance. The variance is computed for each metric per strip structure in the designs. Negative values, in green, mark an improvement and positive values, in red, mark a worsening, p.179

, 185 7.2 Dependencies between two-colouring types and topological aspect in quad meshes

, Condensed results of two-colourable topology finding, p.213

, Presence matrix of the nine strips for the five topologies: 1 if the strip is present, 0 otherwise

, Folding matrix of the nine strips for the ten folded designs: 1 if the strip is present and folded, 0 otherwise, p.224

, Performance matrix of the strain energy in kN.m of the ten folded designs averaged across the different values of amplitude, p.224

, Covariance matrix in kN 2 .m 2 of the energy performance of the folded strips, averaged across all designs, p.224

, Hexagonal shape (Figure 7.28)

230 7.10 Pentagonal shape bis (Figure 7.29b), Pentagonal shape (Figure 7.29a) ,

Rectangular shape with one opening (Figure 7.30), p.230 ,

Rectangular shape with two openings (Figure 7.31), p.230 ,

, Numerical results of the folded shells -part 1, p.231

, Numerical results of the folded shells -part 2, p.232

, Genotype distance d G and phenotype distance d P between a set of three strings and the resulting meshes. The results are symmetric

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, About the author Robin Oval 2019 -University of Cambridge, pp.2016-2019

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