The 3D model of the Church of Longuelo. 

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... term ‘parametric’ was born in the world of mathematics to define all those “equations that express a set of quantities as explicit functions of a number of independent variables, known as ‘parameters’” (Weisstein 2003). Its architectural transposition was introduced by Luigi Moretti during the Forties and indicates a consistent system of elements and relationships among them, which is founded on the construction of topological rather than metrical spaces. The term ‘parametric’ identifies the only invariant of each and every design process with the potential of standardisation in digital design (Aish, 2005). Through parametrics, digital technology is changing into a resource for conceptual design, which is used to formulate problems in a different way and construct resolution tools and strategies in an interactive manner. This paper aims to highlight this phenomenon by means of parametrically redesigning the historical case study of the church of Longuelo, by architect Pino Pizzigoni (1961-1966). Before the introduction of NURBS-based CAD software and optimisation, the design of form-resistant structures was based on the use of either experimental tools (physical form-finding) or analytical surfaces, and architects were challenged in the articulation of spaces from the intrinsic characteristics/rules of structural forms. The presence of an existing project permits a clear comparison between design processes and shows where conceptual design is now taking place. Pino Pizzigoni designed the church of Longuelo, near Bergamo, in Italy, between 1961 and 1966. This is generally considered his most relevant project based on shell structures. His interest for shell and spatial structures started after the Second Post War. First, he built a few prototypes of hyperbolic paraboloids and umbrellas in a field of his own property. Then, he applied that experience to the roof design of some schools and factories around Bergamo (Deregibus and Pugnale, 2010). The church spans over 900 square meters, for 18 meters top height. It is conceived as a double-reflectional symmetrical room made of four identical quarters. Each quarter is made of 5 reinforced concrete hypar shells and a beam frame. Figures 1 and 2 show, respectively, an external and an internal view of the church. 2.1 ORIGINAL DESIGN AND CONCEPTS Pizzigoni’s archive is currently located in the public library of Bergamo “Angelo Mai”. It can be assumed from the documents found that Pizzigoni based the design of the church of Longuelo on three main propositions: Geometrically speaking, four hypar-surfaces are joined together in the shape of a ring and then topologically transformed into a Möbius strip. A fifth hypar is placed between the other ones in order to give strength to the overall configuration. The use of hypar shells offers several spatial configurations for the proposed frame. Moreover, the topology through which he ar- ranged the hypars, allowed the architect to put to practice the concept of Möbius ring as well. Another feature related to the hypar is that the internal space results in a single, smooth surface that echoes the concept of the tent pitched by God, metaphorical / biblical concept described by the gospel of John (Deregibus and Pugnale, 2010). In order to better understand Pizzigoni’s design, a three-dimensional ® model in Rhinoceros of the church has been developed. Free bars of the frame have been explicitly avoided in order to focus on the spatial complexi- ty provided by the shells (Figure 4). Now, by focusing on a single quarter, it is possible to understand how the geometry of the shells works (Figure 5). According to a classical engineering approach, the church would now be op- timised to verify the accuracy of the original structural design and, therefore, the extent of Pizzigoni’s expertise and intuition. Such a process generally starts from a parametric definition of the existing geometry, as it was conceived by the architect - no design is involved, but just a form-improvement of a well-defined structural layout is performed. In this Section, a different approach is shown. The aim is to use parametric design and optimisation to define a computational process of morphogenesis, in which conceptual design is highly involved. 3.1 DESIGN OF THE PARAMETRIC MODEL AND OPTIMISATION As mentioned before, three main concepts led Pizzigoni towards the final design of the church: (1) the use of hyperbolic paraboloids, (2) the Möbius ring and (3) the God Tent. The parametric model of the church is here defined from such concepts rather than referring to the built geometry. This permits the optimisation algorithm to generate new spatial configurations of the church, and therefore it allows morphogenesis to take place. ® ® The model is generated by means of Grasshopper , a Rhinoceros plug- in that allows the definition of parametric systems and their visualisation in real-time. First, the symmetrical and centrical layout of the original church is preserved. The parametric model is therefore based on a single quarter of the church, which is subsequently mirrored twice. Second, each quarter of the original church is composed by five hypars, four of which define a Möbius ring. The topology of such a ring is preserved by the parametric model, but the position of the fifth hypar becomes a design variable. Three different edge conditions are possible for this hypar, as shown in Figure 6. Third, the structural frame is determined by the edges of the hypars, and the geometry of the edges is controlled by acting on the coordinates of eight nodes. Defining the solution domain of such coordinates is a key point (Figure 7). A bounding box that embraces the volume of a church quarter is defined and acts as the overall domain for positioning the eight nodes. Four nodes are further constrained to the upper part of the box to generate a closed roof (R 1 , R 2 , R 3 and R 4 ); two points are retained to the lower part of the quarter to define the connections with the ground (B 1 and B 2 ); the last two points are kept into the middle part of the quarter (M 1 and M 2 ). Defining a parametric model of the church in this manner involves design. Therefore, this is only one of the possible ways to interpret the concepts by Pizzigoni, and has a purely illustrative scope of what ‘parametric thinking’ stands for. Optimisation is here used as a source of inspiration. The aim is to develop the church design through the generation of a wide spectrum of geometries, which perform efficiently in structural, spatial and functional terms. In order to do so, multi-objective optimisation is used. Structural behaviour is the first performance. Nodal vertical displacements of the church frame are calculated by means of the FE solver Karam- ® ba, a Grasshopper plug-in which was developed by Clemens Preisinger with Bollinger+Grohmann (Preisinger, 2013). Karamba is chosen because permits to rapidly connect geometric and structural models within a single ® parametric environment, i.e. Grasshopper . Applied loading refers to the original calculations provided by Pizzigoni (Deregibus and Pugnale, 2010). Spatiality and functionality are then considered together as the second performance criterion. The idea is to calculate the maximum amount of space covered by the structure with no intermediate supports. Without this function, the algorithm would tend to reduce the structural frame size towards unusable dimensions. Galapagos, a ‘black-box’ Genetic Algorithm implemented in Grasshop- per®, is used to perform the optimisation. Genetic Algorithms (GAs) are meta-heuristic search algorithms based on the mechanics of natural selection and natural genetics (Coelho et al., 2014). They provide a robust and flexible tool to solve complex problems and their meta-heuristic way of exploring suitable solutions seems to be particularly helpful in architecture – designers generally benefit from comparing several sub-optimal outcomes rather than converging to a single optimal one. Multi-objective is pursued through the aggregated fitness function F (1). Such a function aims to minimise nodal displacements (2) while the area covered by the shells is maximised (3). For both criteria, solution domains are ...