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Figure 4: Node axis and surface normals The shape operator can be interpreted as a combination of two 1D-scalings of the tangent plane at P, respectively in the directions e; and e2, with factors —k, and -k2. As we will consider structures in which node axes do not coincide with the normal vectors of the envelope surface, we will use the upper case N to designate a node axis, and the lower case n to designate the surface normal, as illustrated in Figure 4. the upper case N to designate a node axis, and the lower

Figure 4 Node axis and surface normals The shape operator can be interpreted as a combination of two 1D-scalings of the tangent plane at P, respectively in the directions e; and e2, with factors —k, and -k2. As we will consider structures in which node axes do not coincide with the normal vectors of the envelope surface, we will use the upper case N to designate a node axis, and the lower case n to designate the surface normal, as illustrated in Figure 4. the upper case N to designate a node axis, and the lower

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Figure 1: An alternative panelization of the SAGE music center with an Archimedean hex-quad-tri tiling. Compared to the original design by F+P, faces are no longer planar, but nodes are still free of geometrical torsion (with 0.5° tolerance), while node axes remain on average within 2° of surface normal (bottom right picture © W. Nicholson). Patterns for gridshells with negligible geometrical torsion at nodes”
Figure 1: An alternative panelization of the SAGE music center with an Archimedean hex-quad-tri tiling. Compared to the original design by F+P, faces are no longer planar, but nodes are still free of geometrical torsion (with 0.5° tolerance), while node axes remain on average within 2° of surface normal (bottom right picture © W. Nicholson). Patterns for gridshells with negligible geometrical torsion at nodes”
Figure 2: Common strategies to deal with geometrical torsion. Top row: strategy #1. Left: Centre Pompidou Metz (©D. Boy de la Tour); Middle: Jay Pritzker Pavilion; Right: MyZeil shopping mall (OR. Mesnil). Bottom left: strategy #2, France Pavilion at Milan World Expo (©Simonin). Bottom right: strategy #3
Figure 2: Common strategies to deal with geometrical torsion. Top row: strategy #1. Left: Centre Pompidou Metz (©D. Boy de la Tour); Middle: Jay Pritzker Pavilion; Right: MyZeil shopping mall (OR. Mesnil). Bottom left: strategy #2, France Pavilion at Milan World Expo (©Simonin). Bottom right: strategy #3
Figure 3: Gauss map of a smooth surface where u = UW 4 + U2e2 is the decomposition of u in the basis of principal curvature directions (e1, e2) of Tp.
Figure 3: Gauss map of a smooth surface where u = UW 4 + U2e2 is the decomposition of u in the basis of principal curvature directions (e1, e2) of Tp.
Figure 5: Visualization of the three strategies to deal with torsion on the linearized Gauss mat 2.2 A sufficient rule for low torsion Strategy 3 gives an interesting trade-off compared to strate- gies 1 and 2. However, the optimization might yield lim- ited improvements, with axes that still deviate significantly from surface normals. This can be observed in Figure 5 (right): as we move away from point A, the distance be- tween n and N (i.e. the angle between axes and normal) keeps increasing, such that high kinks between panels and beams will be unavoidable far away from A. In the ordinary case, there is thus an accumulation of deviation between axes and surface normals. Strategy 3 gives an interesting trade-off compared to strate-
Figure 5: Visualization of the three strategies to deal with torsion on the linearized Gauss mat 2.2 A sufficient rule for low torsion Strategy 3 gives an interesting trade-off compared to strate- gies 1 and 2. However, the optimization might yield lim- ited improvements, with axes that still deviate significantly from surface normals. This can be observed in Figure 5 (right): as we move away from point A, the distance be- tween n and N (i.e. the angle between axes and normal) keeps increasing, such that high kinks between panels and beams will be unavoidable far away from A. In the ordinary case, there is thus an accumulation of deviation between axes and surface normals. Strategy 3 gives an interesting trade-off compared to strate-
Figure 6: Supposing that curvature directions are horizontal and vertical, the Archimedian octo-quad tiling fulfills the rules The most simple torsion-free struc aligned with principal curva of rules. Supposing that curvature direc pattern of Figure 6 also fulfills and vertical, hem: the en and there exi inside the ce he cell boun ranslating the square primi ion is indeed he octo-quad sts a parallel dary (rule 2/). ire pattern can be obtained ive cell ind ure directions — fulfills this se ure — quadrangles ions are horizonta by copying and ashed red (rule 1/), ransforma 1 that results in a horizon The fact tha parallel can colored lines be observed orange, blue, green and pur ransformation: its length changes, but not Regarding rule 2/, we can observe on Figure 6 that finding one value of A implies that we can find any o even though this might result in edges changing orientation far right picture, colors). ion of the pattern al elongation of the transforma- by following the ple) during the its orientation. her values of A,
Figure 6: Supposing that curvature directions are horizontal and vertical, the Archimedian octo-quad tiling fulfills the rules The most simple torsion-free struc aligned with principal curva of rules. Supposing that curvature direc pattern of Figure 6 also fulfills and vertical, hem: the en and there exi inside the ce he cell boun ranslating the square primi ion is indeed he octo-quad sts a parallel dary (rule 2/). ire pattern can be obtained ive cell ind ure directions — fulfills this se ure — quadrangles ions are horizonta by copying and ashed red (rule 1/), ransforma 1 that results in a horizon The fact tha parallel can colored lines be observed orange, blue, green and pur ransformation: its length changes, but not Regarding rule 2/, we can observe on Figure 6 that finding one value of A implies that we can find any o even though this might result in edges changing orientation far right picture, colors). ion of the pattern al elongation of the transforma- by following the ple) during the its orientation. her values of A,
Figure 7: As this octo-quad pattern fullfills the rules, the angle between node axes and surface normals do not increase as we move away fromA
Figure 7: As this octo-quad pattern fullfills the rules, the angle between node axes and surface normals do not increase as we move away fromA
Figure 10: Patterns to be compared. Left: same pattern, aligned or not with principal curvature directions. Middle: Two patterns with same connectivity. One has kinks such that it fulfills the rules. Right: Two Archimedean patterns based on hexagons and triangles, only the left one fulfills the rules As the proposed rule is based on local asymptotic analy- sis, this section studies its validity on a global scale. For this purpose, we perform case studies on meshes covering reference surfaces. We compare numerically three pairs of patterns with similar connectivity and edge lengths, one ful- filling the rules, the others not. We optimize the node axes orientation to obtain torsion-free gridshells and we then
Figure 10: Patterns to be compared. Left: same pattern, aligned or not with principal curvature directions. Middle: Two patterns with same connectivity. One has kinks such that it fulfills the rules. Right: Two Archimedean patterns based on hexagons and triangles, only the left one fulfills the rules As the proposed rule is based on local asymptotic analy- sis, this section studies its validity on a global scale. For this purpose, we perform case studies on meshes covering reference surfaces. We compare numerically three pairs of patterns with similar connectivity and edge lengths, one ful- filling the rules, the others not. We optimize the node axes orientation to obtain torsion-free gridshells and we then
Figure 11: Deviation of the node axes from the surface normals for the six patterns on the first reference surface
Figure 11: Deviation of the node axes from the surface normals for the six patterns on the first reference surface
Figure 12: Deviation of the node axes from the surface normals for the six patterns on the second reference surface
Figure 12: Deviation of the node axes from the surface normals for the six patterns on the second reference surface
Figure 13: A torsion-free octo-quad pattern Int his section, the vast design possibilities offered by pat: terns fulfilling the rules, is shown. Figure 13 presents a sur The face with planar generation steps curvature lines, generated from [16] described in section 3.4 are performed to cover it by an octo-quad pattern. The average deviation of t sho Arc ofa he beams from t have a singularity a gularity at that point: three quadrangles meet at the same point, contrary to the other vertices. The average deviation of node axes is 5.8°. stadium like sha he surface normals is 1.8°. Figure 14 ws a constant mean curvature surface covered by an himedean tri-quad pattern. Because the curvature lines the apex, the pattern also has a sin- Finally, Figure 15 shows the geometry pe based on a Monge surface, covered by a hex-quad pattern. The average deviation of beams from surface normals is 1.9°.
Figure 13: A torsion-free octo-quad pattern Int his section, the vast design possibilities offered by pat: terns fulfilling the rules, is shown. Figure 13 presents a sur The face with planar generation steps curvature lines, generated from [16] described in section 3.4 are performed to cover it by an octo-quad pattern. The average deviation of t sho Arc ofa he beams from t have a singularity a gularity at that point: three quadrangles meet at the same point, contrary to the other vertices. The average deviation of node axes is 5.8°. stadium like sha he surface normals is 1.8°. Figure 14 ws a constant mean curvature surface covered by an himedean tri-quad pattern. Because the curvature lines the apex, the pattern also has a sin- Finally, Figure 15 shows the geometry pe based on a Monge surface, covered by a hex-quad pattern. The average deviation of beams from surface normals is 1.9°.
Figure 15: The geometry of a stadium covered by a hex-quad pattern (4th pattern on Figure 8) Figure 14: A constant mean curvature surface with a singularity
Figure 15: The geometry of a stadium covered by a hex-quad pattern (4th pattern on Figure 8) Figure 14: A constant mean curvature surface with a singularity
Figure 16: Distance between node axis and surface normal for two patterns
Figure 16: Distance between node axis and surface normal for two patterns
Figure 17: Effect of edge length and torsion convergence criterion on the mean deviation angle between node axes and surface normals for a skew quad pattern on a cylinder 5 Conclusion
Figure 17: Effect of edge length and torsion convergence criterion on the mean deviation angle between node axes and surface normals for a skew quad pattern on a cylinder 5 Conclusion
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ArchitectureTorsion Spring
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