. .. Summary-of-contributions, , vol.180, p.183

.. .. Motivations,

. .. Two-colouring-characterisation, 187 7.2.1 Types of two-colouring

. .. Two-colouring-search, 197 7.3.1 Third-colour deletion

. .. Topology-finding,

. .. Summary-of-contributions, , p.226

. .. Appendix, , p.269

. .. Contributions, 271 10.2.3 Mapping and clustering for aided exploration

.. .. Conclusions,


. Alvarez, 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

A. .. , 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

. .. Topologies, 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

;. Bibliography and . Adriaenssens, Shell structures for architecture: form finding and optimization, 2014.

. Adriaenssens, Finding the form of an irregular meshed steel and glass shell based on construction constraints, Journal of Architectural Engineering, vol.18, issue.3, pp.206-213, 2012.

C. Agarwal, M. Agarwal, and J. Cagan, A blend of different tastes: the language of coffeemakers, Environment and Planning B: Planning and Design, vol.25, issue.2, pp.205-226, 1998.

. Aish, Topologic: tools to explore architectural topology, Advances in Architectural, 2018.

. Akleman, Strip sculptures. In 2010 Shape Modeling International Conference, pp.236-240, 2010.

. Akleman, Construction with physical version of quad-edge data structures, Computers & Graphics, vol.58, pp.172-183, 2016.

. Alvarez, Tailored structures, robotic sewing of wooden shells, 2018.

, Robotic Fabrication in Architecture, Art and Design, pp.405-420

A. , Sustainability assessment of a lightweight biomimetic ceiling structure, Bioinspiration & biomimetics, vol.9, issue.1, p.16013, 2014.

R. Avelino and O. Baverel, Structural analysis of gridshells designed from singularities, Proceedings of the Annual Symposium of the International Association for Shell and Spatial Structures, 2017.

. Ayres, Beyond the basket case: a principled approach to the modelling of kagome weave patterns for the fabrication of interlaced lattice structures using straight strips, Advances in Architectural, 2018.

R. Baldock-;-baldock, Structural optimisation in building design practice: case-studies in topology optimisation of bracing systems, 2007.

[. Baldock, Evolving optimized braced steel frameworks for tall buildings using modified pattern search, Computing in Civil Engineering, 2005.

M. R. Barnes-;-barnes, Form finding and analysis of tension structures by dynamic relaxation, International Journal of Space Structures, vol.14, issue.2, pp.89-104, 1999.

O. Baverel, Nexorades: a family of interwoven space structures, 2000.

P. Baverel, . Larsen, O. Baverel, and O. Popovic-larsen, A review of woven structures with focus on reciprocal systemsnexorades, International Journal of Space Structures, vol.26, issue.4, pp.281-288, 2011.

[. Beghini, Connecting architecture and engineering through structural topology optimization. Engineering Structures, vol.59, pp.716-726, 2014.

[. Beirão, Creating specific grammars with generic grammars: towards flexible urban design, Nexus Network Journal, vol.13, issue.1, pp.73-111, 2011.

S. ;. Bendsøe, M. P. Bendsøe, and O. Sigmund, Topology optimization: theory, methods, and applications, 2013.

[. Bhooshan, , 2018.

, Mayavault -A mesh modelling environment for discrete funicular structures, Nexus Network Journal, pp.1-16

[. Bielefeldt, L-system-generated mechanism topology optimization using graph-based interpretation, Journal of Mechanisms and Robotics, vol.11, issue.2, p.20905, 2019.

[. Bielefeldt, Development and validation of a genetic l-system programming framework for topology optimization of multifunctional structures, Computers & Structures, vol.218, pp.152-169, 2019.

R. Bletzinger, K. Bletzinger, and E. Ramm, Form finding of shells by structural optimization. Engineering with computers, vol.9, pp.27-35, 1993.

R. Bletzinger, K. Bletzinger, and E. Ramm, A general finite element approach to the form finding of tensile structures by the updated reference strategy, International Journal of Space Structures, vol.14, issue.2, pp.131-145, 1999.

[. Block, . Ochsendorf, P. Block, and J. Ochsendorf, Thrust Network Analysis: A new methodology for three-dimensional equilibrium, Journal of the International Association for Shell and Spatial Structures, vol.48, issue.3, pp.167-173, 2007.

H. Blum-;-blum, A transformation for extracting new descriptors of shape. Models for Perception of Speech and Visual Forms, pp.362-380, 1967.

[. Bobenko, H. Bobenko, A. I. Huhnen-venedey, and E. , Curvature line parametrized surfaces and orthogonal coordinate systems: discretization with dupin cyclides, Geometriae Dedicata, vol.159, issue.1, pp.207-237, 2014.

. Bommes, , 2009.

, Mixed-integer quadrangulation, Association for Computing Machinery Transactions On Graphics, vol.28, issue.3, p.77

A. Borgart, New challenges for the structural morphology group, Journal of the International Association for Shell and Spatial Structures, vol.51, issue.3, pp.183-189, 2010.

[. Botsch, Polygon mesh processing, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00538098

[. Bouaziz, Projective dynamics: fusing constraint projections for fast simulation, Association for Computing Machinery Transactions on Graphics, vol.33, issue.4, p.154, 2014.

[. Brocato, . Mondardini, M. Brocato, L. ;. Mondardini, L. Hesselgren et al., Geometric methods and computational mechanics for the design of stone domes based on Abeille's bond, Advances in Architectural Geometry 2010, pp.149-162, 2010.

[. Brown, , 2015.

, Multi-objective optimization for diversity and performance in conceptual structural design, Proceedings of the Annual Symposium of the International Association for Shell and Spatial Structures, 2015.

J. Cagan, Engineering shape grammars: where we have been and where we are going, Formal engineering design synthesis, pp.65-92, 2001.

[. Caigui, Checkerboard patterns with black rectangles. Association for Computing Machinery Transactions on Graphics, 2019.

C. Calladine, Buckminster fuller's "tensegrity" structures and clerk maxwell's rules for the construction of stiff frames, International journal of solids and structures, vol.14, issue.2, pp.161-172, 1978.

[. Campen, Dual loops meshing: quality quad layouts on manifolds, Association for Computing Machinery Transactions on Graphics, vol.31, issue.4, p.110, 2012.

[. Campen, Quantized global parametrization, Association for Computing Machinery Transactions on Graphics, vol.34, issue.6, p.192, 2015.

. Campen, M. Kobbelt-;-campen, and L. Kobbelt, Dual strip weaving: Interactive design of quad layouts using elastica strips. Association for Computing Machinery Transactions on Graphics, vol.33, p.183, 2014.

C. ;. Catmull, E. Catmull, and J. Clark, Recursively generated B-spline surfaces on arbitrary topological meshes, Computer-Aided Design, vol.10, issue.6, pp.350-355, 1978.

P. L. Chew, Constrained Delaunay triangulations. Algorithmica, vol.4, pp.97-108, 1989.

N. Chomsky, Three models for the description of language, IRE Transactions on information theory, vol.2, issue.3, pp.113-124, 1956.

N. Chomsky, On certain formal properties of grammars, Information and control, vol.2, issue.2, pp.137-167, 1959.

R. P. Clune, Algorithm selection in structural optimization, 2013.

[. Conway, The symmetries of things, 2016.

[. Cordella, An improved algorithm for matching large graphs, 3rd IAPR-TC15 workshop on graph-based representations in pattern recognition, pp.149-159, 2001.

D. P. Dailey-;-dailey, Uniqueness of colorability and colorability of planar 4-regular graphs are np-complete, Discrete Mathematics, vol.30, issue.3, pp.289-293, 1980.

. Danhaive, R. A. Mueller-;-danhaive, and C. T. Mueller, Combining parametric modeling and interactive optimization for highperformance and creative structural design, Proceedings of the Annual Symposium of the International Association for Shell and Spatial Structures, 2015.

[. Daniels, Quadrilateral mesh simplification. Association for Computing Machinery Transactions on Graphics, vol.27, p.148, 2008.

I. I. Daniels, Localized quadrilateral coarsening, Computer Graphics Forum, vol.28, issue.5, pp.1437-1444, 2009.

R. Dawkins, The Blind Watchmaker, 1987.

K. Deb-;-deb, Multi-objective optimization using evolutionary algorithms, vol.16, 2001.

. Del-coso, Mixing numerical and categorical data in a self-organizing map by means of frequency neurons, Applied Soft Computing, vol.36, pp.246-254, 2015.

[. Deleuran, Exploratory topology modelling of form-active hybrid structures. Procedia Engineering, vol.155, pp.71-80, 2016.

. Deuss, Shapeop-a robust and extensible geometric modelling paradigm, Modelling Behaviour, pp.505-515, 2015.

[. Douthe, Isoradial meshes: Covering elastic gridshells with planar facets, Automation in Construction, vol.83, pp.222-236, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01622864

. [du-peloux, From shape to shell: a design tool to materialize freeform shapes using gridshell structures, Rethinking Prototyping: Design Modelling Symposium, 2013.

. [du-peloux, The ephemeral cathedral of Créteil: a 350m2 lightweight gridshell structure made of 2 kilometers of GFRP tubes, Conférence Internationale de Géotechniques, des Ouvrages et Structures 2015: Innovations in Construction, 2015.

J. P. Duarte-;-duarte, A discursive grammar for customizing mass housing: the case of siza's houses at malagueira. Automation in construction, vol.14, pp.265-275, 2005.

[. Duarte, An urban grammar for the medina of marrakech, Design computing and cognition'06, pp.483-502, 2006.

[. Duballet, Space truss masonry walls with robotic mortar extrusion, vol.18, pp.41-47, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02089858

H. Dürr, Seilnetze-Planung, Berechnung, Ausführung und Werkplanung. Stahlbau, vol.69, issue.8, pp.585-594, 2000.

D. Costa, E. C. Costa, and J. P. Duarte, Mass customization of ceramic tableware through digital technology, Green Design, Materials and Manufacturing Processes, pp.467-471, 2013.

. Echenagucia, T. M. Block-;-echenagucia, and P. Block, Acoustic optimization of funicular shells, Proceedings of the Annual Symposium of the International Association for Shell and Spatial Structures, 2015.

E. , D. Eloy, S. Duarte, and J. P. , A transformation grammar for housing rehabilitation, Nexus Network Journal, vol.13, issue.1, pp.49-71, 2011.

D. Eppstein, Lecture notes in ics 161: Design and analysis of algorithms, 2007.

C. Ertelt and K. Shea, Generative design and cnc fabrication using shape grammars, ASME 2008 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, pp.25-34, 2009.

. Estrin, Topological interlocking as a material design concept, Materials Science and Engineering: C, vol.31, issue.6, pp.1189-1194, 2011.

L. Euler, Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum Petropolitanae, pp.128-140, 1741.

E. Fedorov, Simmetrija na ploskosti, 1891.

Z. Obshchestva, Proceedings of the Imperial St. Petersburg Mineralogical Society, vol.28, pp.245-291

U. Flemming, More than the sum of parts: the grammar of queen anne houses, Environment and Planning B: Planning and Design, vol.14, issue.3, pp.323-350, 1987.

[. Fogg, Enhanced medial-axis-based block-structured meshing in 2-D, Computer-Aided Design, vol.72, pp.87-101, 2016.

[. Fogg, Singularities in structured meshes and crossfields, Computer-Aided Design, vol.105, pp.11-25, 2018.

, Apps for rhino and grasshopper, p.4, 2019.

W. Francis, G. K. Francis, and J. R. Weeks, Conway's zip proof. The American mathematical monthly, vol.106, pp.393-399, 1999.

[. Fuhrimann, Data-driven design: Exploring new structural forms using machine learning and graphic statics, Proceedings of the Annual Symposium of the International Association for Shell and Spatial Structures, 2018.

S. Garcia, Classifications of shape grammars, Design Computing and Cognition'16, pp.229-248, 2017.

P. Geyer-;-geyer, Multidisciplinary grammars supporting design optimization of buildings, vol.18, pp.197-216, 2008.

É. Ghys, Topologie des feuilles génériques, Annals of Mathematics, pp.387-422, 1995.

[. Glymph, A parametric strategy for free-form glass structures using quadrilateral planar facets, Automation in Construction, vol.13, issue.2, pp.187-202, 2004.

D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, 1989.

[. Goldman, Turtle geometry in computer graphics and computer-aided design, Computer-Aided Design, vol.36, issue.14, pp.1471-1482, 2004.

. Gondran, M. Minoux-;-gondran, and M. Minoux, Graphs and algorithms, 1984.
URL : https://hal.archives-ouvertes.fr/hal-01304880

. Grünbaum, B. Shephard-;-grünbaum, and G. C. Shephard, Tilings and patterns, 1987.

[. Hagberg, Exploring network structure, dynamics, and function using networkx, 2008.

[. Halpern, The ribbed floor slab systems of Pier Luigi Nervi, Journal of the International Association for Shell and Spatial Structures, issue.23, pp.1-7, 2013.

R. W. Hamming, Error detecting and error correcting codes. The Bell system technical journal, vol.29, pp.147-160, 1950.

M. Hansmeyer, B. ;. Dillenburger, P. Janssen, S. Roudavski, and B. Tunçer, Mesh grammars, Conference on Computer-Aided Architectural Design Research in, pp.821-829, 2013.

J. Harding-;-harding, S. Adriaenssens, F. Gramazio, M. Kohler, A. Menges et al., Dimensionality reduction for parametric design exploration, Advances in Architectural Geometry, pp.274-87, 2016.

B. Harding, J. Harding, and C. Brandt-olsen, Biomorpher: Interactive evolution for parametric design, International Journal of Architectural Computing, vol.16, issue.2, pp.144-163, 2018.

[. Harding, Thinking topologically at early stage parametric design, Advances in Architectural Geometry, 2012.

, Meta-parametric design. Design Studies, vol.52, pp.73-95

[. Hartl, Additive topological optimization of muscular-skeletal structures via genetic l-system programming, 24th AIAA/AHS Adaptive Structures Conference, p.1569, 2016.

L. Gilbert-;-he and M. Gilbert, Automatic rationalization of yield-line patterns identified using discontinuity layout optimization, International Journal of Solids and Structures, vol.84, pp.27-39, 2016.

H. Hecker, Der Hörsaal des Zoologischen Instituts der, vol.25, pp.49-52, 1969.

L. Heisserman, Generative geometric design, IEEE Computer Graphics and Applications, vol.14, issue.2, pp.37-45, 1994.

J. Heyman, The stone skeleton: structural engineering of masonry architecture, 1997.

[. Hojjat, The vertex morphing method for node-based shape optimization, Computer Methods in Applied Mechanics and Engineering, vol.268, pp.494-513, 2014.

C. Hsu, Generalizing self-organizing map for categorical data, IEEE transactions on Neural Networks, vol.17, issue.2, pp.294-304, 2006.

[. Hu, Pattern mapping with quad-pattern-coverable quad-meshes, Computers & Graphics, vol.36, issue.5, pp.455-465, 2012.

T. Hull-;-hull, On the mathematics of flat origamis, Congressus numerantium, pp.215-224, 1994.

[. Jakob, Instant field-aligned meshes. Association for Computing Machinery Transaction on Graphics, vol.34, pp.189-190, 2015.

[. Jiang, Polyhedral patterns. Association for Computing Machinery Transactions on Graphics, vol.34, p.172, 2015.

S. G. Johnson, The nlopt nonlinear-optimization package, 2015.

[. Kälberer, , 2007.

, Quadcover-surface parameterization using branched coverings, Computer graphics forum, vol.26, issue.3, pp.375-384

[. Kilian, Material-minimizing forms and structures. Association for Computing, Machinery Transactions on Graphics, vol.36, issue.6, p.173, 2017.

T. Knight and G. Stiny, Making grammars: From computing with shapes to computing with things, Design Studies, vol.41, pp.8-28, 2015.

T. W. Knight, The generation of hepplewhite-style chair-back designs. Environment and planning B: planning and design, vol.7, pp.227-238, 1980.

. Kobayashi and M. H. Kobayashi, On a biologically inspired topology optimization method, Communications in Nonlinear Science and Numerical Simulation, vol.15, issue.3, pp.787-802, 2010.

T. Kohonen, Essentials of the self-organizing map, Neural networks, vol.37, pp.52-65, 2013.

. Konakovi?-lukovi?, Computational design of deployable auxetic shells, Advances in Architectural, 2018.

. Konakovi?-lukovi?, Rapid deployment of curved surfaces via programmable auxetics, Association for Computing Machinery Transactions on Graphics, vol.37, issue.4, p.106, 2018.

E. Koning, H. Koning, and J. Eizenberg, The language of the prairie: Frank lloyd wright's prairie houses. Environment and planning B: planning and design, vol.8, pp.295-323, 1981.

[. Koronaki, Layout optimization of space frame structures, Proceedings of the Annual Symposium of the International Association for Shell and Spatial Structures, 2017.

[. Längst, The potentials of isogeometric analysis methods in integrated design processes, Proceedings of the Annual Symposium of the International Association for Shell and Spatial Structures, 2017.

G. Leach, Improving worst-case optimal delaunay triangulation algorithms, 4th Canadian Conference on Computational Geometry, vol.2, p.15, 1992.

A. Lebée-;-lebée, From folds to structures, a review, International Journal of Space Structures, vol.30, issue.2, pp.55-74, 2015.

. Lebée, A. Sab-;-lebée, and K. Sab, Homogenization of a space frame as a thick plate: Application of the bending-gradient theory to a beam lattice, Computers & Structures, vol.127, pp.88-101, 2013.

[. Lee, Formfinding explorations through geometric transformations and modifications of force polyhedrons, Proceedings of the Annual Symposium of the International Association for Shell and Spatial Structures, 2016.

[. Lee, Automatic generation of diverse equilibrium structures through shape grammars and graphic statics, International Journal of Space Structures, vol.31, issue.2-4, pp.147-164, 2016.

C. F. Leroy, Traité de stéréotomie: comprenant les applications de la géométrie descriptive à la théorie des ombres, la perspective linéaire, la gnomonique, la coupe des pierres et la charpente, 1890.

V. I. Levenshtein, Binary codes capable of correcting deletions, insertions, and reversals. Soviet physics doklady, vol.10, pp.707-710, 1966.

[. Li, A shape grammar for teaching the architectural style of the Yingzao fashi, 2001.

[. Liew, Optimising the load path of compression-only thrust networks through independent sets. Structural and Multidisciplinary Optimization, vol.60, pp.231-244, 2019.

[. Liew, Load-path optimisation of funicular networks, Meccanica, vol.53, issue.1-2, pp.279-294, 2018.

A. Lindenmayer, Mathematical models for cellular interactions in development i. filaments with one-sided inputs, Journal of theoretical biology, vol.18, issue.3, pp.280-299, 1968.

[. Liu, Geometric modeling with conical meshes and developable surfaces, Association for Computing Machinery Transactions on Graphics, vol.25, issue.3, pp.681-689, 2006.

W. ;. Liu, Y. Liu, and W. Wang, On vertex offsets of polyhedral surfaces, Proceedings of the Advances in Architectural Geometry, pp.61-64, 2008.

[. Liu, General planar quadrilateral mesh design using conjugate direction field, Association for Computing Machinery Transactions on Graphics, vol.30, issue.6, p.140, 2011.

, Smooth subdivision surfaces based on triangles, 1987.

[. Lyon, Parametrization quantization with free boundaries for trimmed quad meshing, 2019.

A. [maia, Topology exploration of compression-only networks for form and force optimisation, 2020.

W. ;. Malek, S. Malek, and C. Williams, Structural implications of using cairo tiling and hexagons in gridshells, Proceedings of the Annual Symposium of the International Association for Shell and Spatial Structures, 2013.

W. ;. Malek, S. Malek, and C. J. Williams, The equilibrium of corrugated plates and shells, Nexus Network Journal, 1988.

[. Marquis, Building elastic gridshells from patches, Proceedings of the Annual Symposium of the International Association for Shell and Spatial Structures, 2017.

A. G. Martin-;-martin, A basketmaker's approach to structural morphology, Proceedings of the Annual Symposium of the International Association for Shell and Spatial Structures, 2015.

Y. Masson-;-masson, Existence et construction de réseaux de Chebyshev avec singularités et application aux gridshells, 2017.

. Maxwell and J. C. Maxwell, Xlv. on reciprocal figures and diagrams of forces. The London, Edinburgh, and Dublin Philosophical Magazine and, Journal of Science, vol.27, issue.182, pp.250-261, 1864.

[. Mccormack, Speaking the buick language: capturing, understanding, and exploring brand identity with shape grammars. Design studies, vol.25, pp.1-29, 2004.

. [méndez-echenagucia, A cable-net and fabric formwork system for the construction of concrete shells: design, fabrication and construction of a full scale prototype, 2018.

R. Mesnil-;-mesnil, A re-parameterization approach for the construction of domes with planar facets, Journal of the International Association for Shell and Spatial Structures, vol.59, issue.4, pp.286-295, 2018.
URL : https://hal.archives-ouvertes.fr/hal-02002854

[. Mesnil, Structural morphology and performance of plated structures with planar quadrilateral facets, Journal of the International Association for Shell and Spatial Structures, issue.1, p.58, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01510086

[. Mesnil, Nonstandard patterns for gridshell structures: Fabrication and structural optimization, Journal of the International Association for Shell and Spatial Structures, vol.58, issue.4, pp.277-286, 2017.

[. Mesnil, Form finding of nexorades using the translations method. Automation in Construction, vol.95, pp.142-154, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01899086

[. Mesnil, Generalised cyclidic nets for shape modelling in architecture, International Journal of Architectural Computing, vol.15, issue.2, pp.148-168, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01520152

[. Mesnil, Linear buckling of quadrangular and kagome gridshells: a comparative assessment. Engineering Structures, vol.132, pp.337-348, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01510047

[. Mesnil, Marionette meshes: modelling free-form architecture with planar facets, International Journal of Space Structures, vol.32, issue.3-4, pp.184-198, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01616999

[. Mesnil, Fabrication-aware shape parametrisation for the structural optimisation of shell structures, Engineering Structures, vol.176, pp.569-584, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01899204

[. Meyer, Discrete differential-geometry operators for triangulated 2-manifolds, Visualization and mathematics III, pp.35-57, 2003.

A. G. Michell-;-michell, Lviii. The limits of economy of material in frame-structures. The London, Edinburgh, and Dublin Philosophical Magazine, Journal of Science, vol.8, issue.47, pp.589-597, 1904.

W. J. Mitchell-;-mitchell, Functional grammars: an introduction, Reality and Virtual Reality: Association for Computer Aided Design in Architecture Conference Proceedings, pp.167-176, 1991.

[. Motamedi, Supportless 3d printing of shells: Adaptation of ancient vaulting techniques to digital fabrication, Impact: Design with all Senses: Design Modelling Symposium, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02119439

R. Motro and B. Maurin, Bernard laffaille, nicolas esquillan, two french pioneers, IASS-IABSE Symposium: Taller, Longer, Lighter, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00857310

O. Mueller, C. T. Mueller, and J. A. Ochsendorf, Combining structural performance and designer preferences in evolutionary design space exploration, Automation in Construction, vol.52, pp.70-82, 2015.

[. Nasri, Filling nsided regions by quad meshes for subdivision surfaces, Computer Graphics Forum, vol.28, issue.6, pp.1644-1658, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00867354

G. Navarro, A guided tour to approximate string matching, vol.33, pp.31-88, 2001.

P. Nervi, Aesthetics and technology in building. Charles Eliot Norton lectures, 1965.

H. Nooshin, Algebraic representation and processing of structural configurations, Computers & Structures, vol.5, issue.2-3, pp.119-130, 1975.

H. Nooshin, Formex configuration processing: A young branch of knowledge, International Journal of Space Structures, vol.32, issue.3-4, pp.136-148, 2017.

. Norman, Morphing of curved corrugated shells, International Journal of Solids and Structures, vol.46, issue.7-8, pp.1624-1633, 2009.

W. ;. Nsugbe, E. Nsugbe, and C. Williams, The generation of bone-like forms using analytic functions of a complex variable. Engineering structures, vol.23, pp.22-28, 2001.

[. Ohlbrock, Constraint-driven design with combinatorial equilibrium modelling, Proceedings of the Annual Symposium of the International Association for Shell and Spatial Structures, 2017.

[. Otto, IL 10 Gitterschalen/Grid Shells: Bericht über das japanischdeutsche Forschungsprojekt STI, 1973.

R. Oval, Design exploration of architectural geometries and structural performance for sports arenas based on som-clustering and structural performance simulation, Structures and Architecture-Bridging the Gap and Crossing Borders: Proceedings of the Fourth International Conference on Structures and Architecture (ICSA 2019), 2017.

[. Panozzo, Designing unreinforced masonry models. Association for Computing, Machinery Transactions on Graphics, vol.32, issue.4, p.91, 2013.

[. Pedro, H. C. Kobayashi-;-pedro, and M. H. Kobayashi, On a cellular division method for topology optimization, International Journal for Numerical Methods in Engineering, vol.88, issue.11, pp.1175-1197, 2011.

W. Pele, O. Pele, and M. Werman, The quadraticchi histogram distance family, European conference on computer vision, pp.749-762, 2010.

S. Pellegrino-;-pellegrino, Structural computations with the singular value decomposition of the equilibrium matrix, International Journal of Solids and Structures, vol.30, issue.21, pp.3025-3035, 1993.

D. Pellis, H. Pottmann, L. Hesselgren, A. Kilian, O. Sorkine-hornung et al., Aligning principal stress and curvature directions, Advances in Architectural, 2018.

[. Peng, Exploring quadrangulations. Association for Computing in Machinery, Transactions on Graphics, vol.33, issue.1, p.12, 2014.

[. Peng, Association for Computing Machinery Transactions on Graphics, Kangaroo: form finding with computational physics. Architectural Design, vol.30, pp.136-137, 2011.

G. Polya, Xii. über die analogie der kristallsymmetrie in der ebene, Zeitschrift für Kristallographie-Crystalline Materials, vol.60, issue.1-6, pp.278-282, 1924.

[. Popescu, Automated generation of knit patterns for nondevelopable surfaces, Humanizing Digital Reality: Design Modelling Symposium, pp.271-284, 2017.

[. Pottmann, Architectural Geometry, 2007.

[. Pottmann, Geometry of multi-layer freeform structures for architecture, Association for Computing Machinery Transactions on Graphics, vol.26, issue.3, p.65, 2007.

M. J. Powell-;-powell, A direct search optimization method that models the objective and constraint functions by linear interpolation, Advances in optimization and numerical analysis, pp.51-67, 1994.

M. J. Powell-;-powell, Direct search algorithms for optimization calculations, Acta numerica, vol.7, pp.287-336, 1998.

[. Prado, Elytra filament pavilion: Robotic filament winding for structural composite building systems, 2017.

C. Preisinger, Linking structure and parametric geometry, Architectural Design, vol.83, issue.2, pp.110-113, 2013.

M. Prishlyak, A. O. Prishlyak, and K. I. Mischenko, Classification of noncompact surfaces with boundary, Methods of Functional Analysis and Topology, vol.13, issue.1, pp.62-66, 2007.

. Prusinkiewicz, P. Lindenmayer-;-prusinkiewicz, and A. Lindenmayer, The algorithmic beauty of plants, 2012.

A. Requicha-;-requicha, Mathematical models of rigid solid objects, 1977.

I. Richards, On the classification of noncompact surfaces, Transactions of the American Mathematical Society, vol.106, issue.2, pp.259-269, 1963.

D. Rigby-;-rigby, Topmaker: a technique for automatic multiblock topology generation using the medial axis, 4th Joint Fluids Summer Engineering Conference, pp.1991-1997, 2003.

M. Rippmann and P. Block, Funicular shell design exploration, Proceedings of the 33rd Annual Conference of the ACADIA, 2013.

M. Rippmann and P. Block, Computational tessellation of freeform, cut-stone vaults, Nexus Network Journal, vol.20, issue.3, pp.545-566, 2018.

. [rippmann, Interactive vault design, International Journal of Space Structures, vol.27, issue.4, pp.219-230, 2012.

. [rippmann, Design, fabrication and testing of discrete 3d sand-printed floor prototypes, Materials Today Communications, vol.15, pp.254-259, 2018.

. [rippmann, The Armadillo Vault: computational design and digital fabrication of a freeform stone shell, Advances in Architectural Geometry, pp.344-363, 2016.

[. Ronald, Representational redundancy in evolutionary algorithms, Proceedings of 1995 IEEE International Conference on Evolutionary Computation, vol.2, pp.631-636, 1995.

T. Rowland, G. I. Rozvany, and W. Prager, Optimal design of partially discretized grillages, Journal of the Mechanics and Physics of Solids, vol.24, issue.2-3, pp.125-136, 1976.

S. [ruiz-gironès, E. Ramos-;-ruiz-gironès, and J. Sarrate-ramos, Automatic generation of quadrilateral structured meshes using linear programming and transfinite interpolation, 6th Workshop on Numerical Methods in Applied Science and Engineering, 2007.

[. Saha, A survey on skeletonization algorithms and their applications, Pattern Recognition Letters, vol.76, pp.3-12, 2016.

J. Sakarovitch, Gaspard monge founder of "constructive geometry, Proceedings of the Third International Congress on Construction History, pp.1293-1299, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00712704

. Saldana-ochoa, Beyond typology, beyond optimization, Proceedings of the Annual Symposium of the International Association for Shell and Spatial Structures, 2019.

L. Sass, A wood frame grammar: A generative system for digital fabrication, International Journal of Architectural Computing, vol.4, issue.1, pp.51-67, 2006.

H. Schek, The force density method for form finding and computation of general networks, Computer Methods in Applied Mechanics and Engineering, vol.3, issue.1, pp.115-134, 1974.

A. Schiftner, J. ;. Balzer, L. Hesselgren, M. Pauly, H. Pottmann et al., Staticssensitive layout of planar quadrilateral meshes, Proceedings of the Advances in Architectural Geometry 2010, pp.221-236, 2010.

[. Schiftner, Architectural geometry from research to practice: the eiffel tower pavilions, Advances in Architectural Geometry 2012, pp.213-228, 2012.

I. J. Schlaich and I. H. Schober, Freeform glass roofs, Structures Congress 2005: Metropolis and Beyond, 2005.

J. Schlaich and K. Schafer, Design and detailing of structural concrete using strut-and-tie models, Structural Engineer, vol.69, issue.6, pp.113-125, 1991.

J. Schlaich and H. Schober, Glass roof for the Hippo House at the Berlin Zoo, Structural Engineering International, vol.7, issue.4, pp.252-254, 1997.

[. Schlaich, New trade fair in Milan -grid topology and structural behaviour of a freeformed glass-covered surface, International Journal of Space Structures, vol.20, issue.1, pp.1-14, 2005.

M. Schlaich, Shell bridges-and a new specimen made of stainless steel, Journal of the International Association for Shell and Spatial Structures, vol.59, issue.3, pp.215-224, 2018.

H. Schober, Transparent shells: Form, topology, structure, 2015.

J. Schober, H. Schober, and S. Justiz, Cabot Circus, Bristol -Ebene Vierecknetze für freigeformte Glasdächer, Stahlbau, vol.81, issue.S1, pp.28-42, 2012.

A. Schrijver, Theory of linear and integer programming, 1998.

H. Seifert and W. Threlfall, Lehrbuch der topologie, 1934.

C. Shea, K. Shea, and J. Cagan, Innovative dome design: applying geodesic patterns with shape annealing. Artificial Intelligence for Engineering Design, Analysis and Manufacturing, vol.11, issue.5, pp.379-394, 1997.

C. Shea, K. Shea, and J. Cagan, The design of novel roof trusses with shape annealing: assessing the ability of a computational method in aiding structural designers with varying design intent, Design Studies, vol.20, issue.1, pp.3-23, 1999.

[. Shea, A shape annealing approach to optimal truss design with dynamic grouping of members, Journal of Mechanical Design, vol.119, issue.3, pp.388-394, 1997.

[. Sheffer, ;. De-sturler, A. Sheffer, and E. De-sturler, Parameterization of faceted surfaces for meshing using angle-based flattening, Engineering with computers, vol.17, issue.3, pp.326-337, 2001.

P. Shepherd and W. Pearson, Topology optimisation of algorithmically generated space frames, Proceedings of the Annual Symposium of the International Association for Shell and Spatial Structures, 2013.

P. Shepherd and P. Richens, The case for subdivision surfaces in building design, Journal of the International Association for Shell and Spatial Structures, vol.53, issue.4, pp.237-245, 2012.

[. Sischka, Die Überdachung des Great Court im British Museum in London, Stahlbau, vol.70, issue.7, pp.492-502, 2001.

. Sonntag, Biomimetic timber shells made of bending-active segments, International Journal of Space Structures, vol.32, issue.3-4, pp.149-159, 2017.

[. Soriano, Topologically based curvature in thin elastic shell networks, Textiles composites and inflatable structures VII: proceedings of the VII International Conference on Textile Composites and Inflatable Structures, pp.167-176, 2015.

[. Spadea, Wound frp shear reinforcement for concrete structures, Journal of Composites for Construction, vol.21, issue.5, p.4017026, 2017.

V. Srinivasan, Modeling high-genus surfaces, Texas A&M University. Accessed on, vol.22, issue.10, 2004.

[. Stanford, Simultaneous topology optimization of membrane wings and their compliant flapping mechanisms, AIAA journal, vol.51, issue.6, pp.1431-1441, 2013.

[. Stephan, Reticulated structures on free-form surfaces, Stahlbau, vol.73, issue.8, pp.562-572, 2004.

G. Stiny, Shape: talking about seeing and doing, 2006.

G. Stiny and J. Gips, Shape grammars and the generative specification of painting and sculpture, Proceedings of the Congress International Federation for Information Processing, pp.1460-1465, 1971.

M. ;. Stiny, G. Stiny, and W. J. Mitchell, The palladian grammar. Environment and planning B: Planning and design, vol.5, pp.5-18, 1978.

W. Stitic, A. Stitic, and Y. Weinand, Timber folded plate structures-topological and structural considerations, vol.30, pp.169-177, 2015.

Y. B. Suris and A. I. Bobenko, Discrete differential geometry: Integrable structure, American Mathematical Society, 2008.

S. Suzuki and J. Knippers, The design implications of form-finding with dynamic topologies, Humanizing Digital Reality: Design Modelling Symposium, vol.50, pp.173-179, 2009.

[. Takayama, Pattern-based quadrangulation for N-sided patches, Proceedings of the Symposium on Geometry Processing, pp.177-184, 2014.

[. Takezawa, Fabrication of freeform objects by principal strips. Association for Computing in Machinery Transactions on Graphics, vol.35, p.225, 2016.

[. Tarini, Practical quad mesh simplification, Computer Graphics Forum, vol.29, issue.2, pp.407-418, 2010.

. Turrin, Design explorations of performance driven geometry in architectural design using parametric modeling and genetic algorithms, Advanced Engineering Informatics, vol.25, issue.4, pp.656-675, 2011.

P. J. Van-laarhoven, E. H. Aarts, T. Van-mele, and P. Block, Simulated annealing: Theory and applications, Computer-Aided Design, vol.53, pp.104-116, 1987.

. Van-mele, compas: A framework for computational research in architecture and structures, 2017.

. Van-mele, Best-fit thrust network analysis -Rationalisation of freeform meshes, Shell Structures for Architecture: Form Finding and Optimization, chapter 13, vol.131, pp.755-768, 2014.

[. Wagner, R. A. Fischer-;-wagner, and M. J. Fischer, The string-to-string correction problem, Journal of the Association for Computing Machinery, vol.21, issue.1, pp.168-173, 1974.

J. Wallner and H. Pottmann, Geometric computing for freeform architecture, Journal of Mathematics in Industry, vol.1, issue.1, p.4, 2011.

P. Welsh, D. J. Welsh, and M. B. Powell, An upper bound for the chromatic number of a graph and its application to timetabling problems, The Computer Journal, vol.10, issue.1, pp.85-86, 1967.

C. J. Williams, The analytic and numerical definition of the geometry of the British Museum Great Court Roof, Mathematics & Design, pp.434-440, 2001.

;. Williams and C. J. Williams, Patterns on a surface: the reconciliation of the circle and the square, Nexus Network Journal, vol.13, issue.2, pp.281-295, 2011.

[. Winslow, , 2010.

, Multi-objective optimization of free-form grid structures. Structural and multidisciplinary optimization, vol.40, p.257

Y. Yamaguchi, Classification of boundary representations for manifold and non-manifold topology, Product Modeling for Computer Integrated Design and Manufacture, pp.104-115, 1997.

[. Yamaguchi, Y. Kimura-;-yamaguchi, and F. Kimura, Nonmanifold topology based on coupling entities, IEEE Computer Graphics and Applications, vol.15, issue.1, pp.42-50, 1995.

[. Zadravec, Designing quad-dominant meshes with planar faces, Computer Graphics Forum, vol.29, issue.5, pp.1671-1679, 2010.

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

. Bollinger-+-grohmann, , 2013.

, École des Ponts & Chaussées, France Diplôme d'ingénieur, pp.2010-2012