4. UNSTRUCTURED GRIDS AND STRUCTURAL COMPLEXITY
4.3. Tessellation, transverse systems, and unstructured grids in
Therefore, one of the challenges posed by the digital era is to enact an integral building, a Total Architecture, as has been defined by Arup, an architecture that uses interoperability of sciences in the design process, such as, architecture, structure, construction, production, services, assembly, etc. etc. The related computational challenge is to create software tools, which are also integral. Not only because they consider many aspects, but also because they are integrated and they can also communicate with each other. Nowadays we may see attempts of integration with related programs, such as BIM (Building Information System). Nevertheless, currently, it mostly applies to a design process that involves from the very beginning a lot of information and stages of a certain building. Another big challenge is to include more automation in the design process, for instance in the form of automated optimisation. However, it is important to cover a number of paradigms and grid types, and in particular unstructured grids that are also the main area of the research interest.
4.3. Tessellation, transverse systems, and unstructured
method that accounts for its potentiality of subdividing space with the less or zero number of remaining unused areas. Specifically, it is a structural arrangement of inherent geometrical stability that finds expression in two (2) or three (3) - dimensional space. Additionally, Lisa Iwamoto defines tessellation as a means of closer to fabrication patterning through a “collection of pieces that fit together without gaps to form a plane or a surface”. Definitely, it is a technique used for many decades, especially in Gothic architecture, mostly as decorative patterns that convey a symbolic language, such as the stain-glass windows or Islamic architecture. In contemporary architecture and with the advent of the digital era it has become more as a digitally enhanced tool for the generation of tiled patterns or mesh patterns respectively because a better modulation and variation for non-standard fabrication is afforded. Nevertheless, we may observe the capability of tessellating techniques as a means or producing forms that involve complex shapes and forms, with double-curved surfaces 56 or polygonal meshes. Thus, we may also employ this technique as a means of digitally representing more smooth geometries and in turn easily fabricated through standardised sheet materials.57 However, the interest here reviews tessellation, in a manner of a structural pattern utilisation, that is structurally stable and arranged in that way of offering self-supporting structures, as firstly applied by Buckminster Fuller.
Thus, short presentation of the types of tessellations, as well as the
56 Double curvature surfaces (or anticlastic surfaces) are used to nearly all tensile or membranes fabric structures, to give stability to the membrane. Double curvature can be established when a surface is curved in two directions.
57 Iwamoto, L., (2009), p. 36.
recent and more alternatives ways that have emerged through computation is presented and developed.
There are many alternatives of subdividing or tessellating space, such as regular tessellation, with standardised joints and length members. Regular tessellation makes use of three primary types of shapes, the triangle, the square, and the hexagon, resulting in triangulated, orthogonal and hexagonal combinations respectively.
Based on those shapes we may observe other related combinations, the semi-regular tessellation that can combine at maximum two types of the above-mentioned shapes and illustrated in the figure above (Figure 15). A list with the combinations included in these categories are presented as followed:
hexagons - triangles 60o-60o-60o
squares – squares
octagon/squares - triangles 45o-90o-45o
squares/triangles - pentagons (semi-regular)
Figure 15: Types of regular grids, composed by the three basic grid-shapes. (Square, triangle, hexagon)
squares/triangles - pentagons (semi-regular)
hexagons/triangles – rhombic
dodecagons/hexagons/squares - triangles 30o-60o-90o
hexagons/triangles - pentagons (semi-regular) left- handed
hexagons/triangles - pentagons (semi-regular) right- handed
hexagons/squares/triangles - four-sided-polygons
dodecagons/triangles - triangles 30o-120o-30o
Additionally, many by-products of such combinations do exist in the bibliography, but due to the fact that they are closer to a regular representation, we no further analysis shall be performed towards this direction, than just to a mention:
Dual Tessellations
Tessellations with regular polygons
Tessellations and Symmetry
Open patterns with regular polygons
Compound Tessellations and Islamic patterns
Parallelogram and Rectangular Tessellations
Triangular Tessellations
Zonogonal Tessellations
Central Tessellations
Concentric Patterns with regular Pentagons
Non-periodic Tessellations
Non-periodic Pattern-generation
Tessellation with Non-convex polygons58
Recently, contemporary architecture has started to explore the adoption of more recent mathematical inventions mainly influenced by the “biological paradigm”, including aperiodic tiling, patterns that have smaller in number or repeated units, but whose arrangements are such that the resulting patterns, unlike orthogonal or hexagonal grids, cannot be superimposed upon themselves through translation59. A very interesting perspective and the one that is relative closer to the way this PhD thesis adopts its methodology is the interpretation of Farshid Moussavi in the “Function of Form”. These systems of grid distortion or variation, as mentioned by Moussavi, are an approach to the transversal system, in terms of a system of cross sections with distinct topological variation. According to Moussavi “(…) in a transversal system, a “base unit” assembles a variety of causes and concerns into a complex supramaterial whole – an amorphic rather than hylomorphic whole; (…)”60. Therefore, F. Moussavi views the system with no specific shape but “embedded with protogeometric properties”, such as load bearing capacities that cross-relate the way tessellation is performed. In addition, tessellation follows the notion
“repeat and vary”, in relation to external stimuli and contingencies, such as the site, the climate, local tools, and technologies. Because
58 Coenders, 2003, pp. 32-43.
59 Burry and Burry, 2010, pp. 77-80.
60 Moussavi F., (2009), p.29.
the base unit is not geometrically fixed, it may constantly vary and mutate when hybridised with other units, into novel and unpredictable forms that are spatially specific and capable at the time to adapt to external concerns. In other words, we may see a hybrid way of approach to a “non-standard” design that takes into account a bottom- up approach, from the properties of the system to the influence and performance of the whole. F. Moussavi also observed and illustrated in a relevant table on how a system, in terms of surface, dome, folded plates, shells etc, and in accordance with a specific tessellation can have a variety of affects and effects in the architectural form and structure. Subsequently, this can suggest the numerous capabilities
Figure 16: Relationship between the types of system, tessellation with the affects that in turn they emerge.
(Moussavi F., (2009)
and complex behaviour the research on grid distortion and transformation is able to offer. (Figure 16)
Nevertheless, from a generative perspective, this area of grid structures is notably referred to as unstructured (irregular) grids.
Unstructured or irregular grids are referring to a grid that is assembled with irregular in size and shape components, being, at the same time, torn up and transformed into space, mimicking natural processes of compiling structural patterns, resulting to varieties and singularities.
Many free-form or form-finding designs involve complex regions that are not easily amenable to pure structured grids. Structured grids may lack the required flexibility and robustness for handling complex surfaces, or the grid cells may become too skewed or twisted.
Therefore, the unstructured grid concept is considered as one of the appropriate solutions to the problem of generating grids in areas with complex shapes. On the contrary, an unstructured grid has irregularly distributed nodes and their cells are not obliged to have a standard shape. Apart from this, the connectivity of neighbouring grid cells is not subject to any restrictions. Thus, unstructured grids provide the most flexible tool for the description of a geometrical shape by a mesh.
However, in practice, unstructured grids for architectural structures are not commonly used. In addition, architects and engineers prefer to adapt the shape of the design in order to find a translational structured grid. The grid generation techniques for unstructured grids fill in here as these techniques try to find, with certain accuracy, flat elements.
However, the elements we try to find could be double-curved but should be of least curvature. Thus, the grid generation techniques of
unstructured grids help to find the elements with least double curvature, concerning a certain accuracy and element size.61
Until very recently, grid structures could be generated in three types: the flat skeletal grids, the curvilinear forms of barrel vaults and the domes 62 with reference to the inventions of the architect Buckminster Fuller on “geodesic domes” and furthermore on equal- shape tessellations of shell structures (regular grid structures) (Figure 17).
The recent developments in computation and computational tools have offered the opportunity to exploit three-dimensional grid
61 Coenders, 2003, p. 300.
62 Wilkinson, 1991, pp. 93-100
Figure 17: Several types of grid configuration in curvilinear forms of barrel vaults and domes (Coenders, 2003)
structures and to experiment with complex generative architectural forms that employ structural complexity.
Therefore, in terms of investigating non-standard architectural forms that exploit structural complexity, which as mentioned above accounts for a revival of an organic generation, it is relevant to focus on the unstructured tessellation organised by the subdivision provided by natural structural patterns that tend to exhibit a greater level of complexity and non-linearity. Such structures can be identified within the following groups (Gibson, Lorna, 1997). Nevertheless, this PhD Thesis focuses in the area of cellular structures and honeycombs enlisted below:
Static Natural Structures
Cellular Structures
Honeycombs
Bone Structures
Branching and tree structures
Tree Structures
Leaf Structures
Skeleton Structures
Web Structures
Sea Shell and radiolaria
These biomorphic and natural processes of subdividing space are inspiring examples for digitally translated forms expressions and
Figure 18: Soap bubbles Array (above) (Pearce, 1990), and Radiolara Shape (below) (Coenders, 2003)
are generated by the use of mathematical and computational three- dimensional techniques, such as Delaunay triangulation, Dirichlet or Voronoi63 tessellations, Weairie-Phelan64 foam structure, Catmull- Clark subdivision, etc,
Specifically, Delaunay triangulation (DT) is a mathematical concept that was named after Boris Delaunay for his work on this topic back in 1934. It refers to mathematical and computational geometry and very recently with the advent of computational architecture to
63 Bowyer, 1981
64 Weaire, 2006
Figure 20: Those points define the cell cores of the Voronoi Diagrams, thus becoming the dual of Delaunay triangulation.
(Coenders, 2003)
Figure 19: The circumcircles pass through 3 endpoint of a triangle. In order to satisfy the Delaunay condition, no other point should be included in those circles. (Coenders, 2003)
geometries involved in contemporary architecture. For the Delaunay triangulation usually, a given set of points or vertices P = {p1, p2, p3,…, pn} is considered. The points are connected together by triangles that pass through the set P. Delaunay triangulation is obtained only when the “Delaunay condition” is satisfied. This indicates the creation of circumcircles that pass through the 3 endpoints of every combination of triangles in the triangulation and becomes valid only when no other point of P is included in this circle. Delaunay triangulation maximises the minimum angles of the triangles and tends to avoid any skinny
triangles, smoothing the geometry and becomes part of the mesh geometry creation. (Figure 19, Figure 20)
Voronoi diagrams (VD) on the other hand, are considered to be the dual of Delaunay triangulation (Figure 21), in the sense that these two concepts share similar generative features. Informally studied back in 1694 from Descartes, and 2 decades later from Dirichlet in 1850, but
Figure 21: Generation of Voronoi Diagrams in relation with its dual Delaunay triangulation. (Coenders, 2003)
they have appeared with additional names such as Theisen Polygons, fundamental domains, metric fundamental polygons. These names have been adopted according to the qualities of the space they are included in. In our case, we examine Voronoi in the Euclidean space.
65The name Voronoi diagram was taken after Georgy Fedoseevich Voronoi (Voronoy), a Russian mathematician. In order to generate a Voronoi diagram, an initial set of points P = {p1, p2, p3,…, pn} in the Euclidean space is also considered, which corresponds to n polytopes or polyhedral regions. These polytopes are also known as Voronoi cells, whilst having n number of Voronoi cell cores. To create a valid Voronoi diagram, the distributions of the initial points are first tested for the Delaunay condition validity. The Voronoi boundaries are generated when the perpendicular bisector points found on the Delaunay edge lines are connected. Subsequently, the Delaunay vertices become the Voronoi cell cores, proving this duality of those two mathematical concepts. Voronoi diagrams generate a structural pattern very similar geometrically to a honeycomb configuration. We may observe depending on the distribution of the initial set of points that Voronoi can give rise to a number of different Voronoi diagrams expressions.
A two-dimensional lattice may generate a hexagonal and thus a regular honeycomb, while a randomly distributed set of points
65 Bowyer, A. (1981)
reproduce a rather randomised Voronoi honeycomb, as seen in the figure below. (Figure 22)
Catmull-Clark subdivision is another recent computational subdividing method that is mainly applied in computer graphics to create more smoothing surfaces. It was initially invented by Edwin Catmull and Jim Clark in 1978 as a generalisation of bi-cubic uniform B-spline surfaces to arbitrary topology. The Catmull-Clark subdivision is generated recursively and similarly, with the previous concepts it also requires an initial set of points or mesh vertices to be generated.
The recursion proceeds as following (Figure 23):
• For every face, a new face point is added.
Figure 22: Randomly Voronoi honeycomb from a randomly distributed initial set of points, from which all points that are closer from a minimum distance were removed. (Gibson, Lorna J., 1997)
• For each edge, a new edge point is also added.
• For each face point, add an edge for every face edge, connecting the face point to each edge point for the face.
• For each original point P, take the average F of all n (recently created) face points for faces touching P, and take the average R of all n edge midpoints for edges touching P, where each edge midpoint is the average of its two endpoint vertices. Move each original point to the point: F+ 2R + (n- 3)/n, which corresponds to the barycentre of P, R and F with respective weights (n-3), 2 and 1.
• Connect each new vertex point to the new edge points of all original edges incident on the original vertex
• Define new faces as enclosed by edges.
Figure 23: The process of Cat-mull Clark subdivision.
The new mesh consists of quadrilaterals, which in general will not be planar. The barycentre formula was created by Catmull and Clark more for aesthetics reasons, rather than mathematical.
Presently, a number of examples are observed, that are dealing with the inner structure of materials observed in nature, whilst their ability to self-organized and rearranged under certain conditions.
Cellular materials, as meticulously explained in Chapter 3, are among the ones that represent the function of natural materials structural organisation, such as the case of bone tissues, the structure of sea creatures, wood and other. Bone, for instance, is characterised as a cellular solid, in the means of an internal structure of cells, voids, and space that is filled with fluids or air, each of which has faces of solid or liquid. Dry or liquid foam is also of this type, as well as honeycomb. In addition, although in the industrial world, polymer cellular foams are extensively used for insulating and packing, the high structural efficiency of cellular materials has only begun to taken into consideration and explored, having a slow development, due to the unawareness of architects and engineers of the engineering design of cellular materials. Nevertheless, what has been in development currently is in the fields of aerospace, maritime and medical purposes, with the composition of new “designed” materials, such as polymers.
The unique and interesting thing about them is that they possess properties, such as being lightweight, flexible and mechanically strong.
This technique of manufacturing new materials by mimicking nature’s self-organisational behaviour can actually produce honeycomb
structure at a molecular scale, but due to these properties, they can be considered to be applied in larger scales, although at the moment remain in research. Nevertheless, all these natural structural patterns, due to their characteristic have a lot of potentialities to produce new structures and systems for advanced architectural engineering. The most famous example of built environment adaptation of one of these systems and probably one of the few implemented as being the structure system itself is the example of “Watercube” in Beijing. The project is taking into consideration the dry foam structure, the Weairie- Phelan model, named after their innovators. (Figure 24)
Figure 24: Wearie-Phelan foam structure.
Finally, there are many techniques available, with all the research activity devoted to automatic grid generation, for the construction of unstructured grids. However, three approaches are widely used. They can be described as point insertion methods based on Delaunay triangulation, and Advancing Front Methods and tree- based methods, such as the Octree approach.