*2.2. Establishing of the Structural Model of the Canopy and Assumptions for the Evolutionary Optimization*

Due to the fact that the geometry of the structure plays a crucial role in any optimization problem, the scripts developed to achieve the geometric forms of the roof structures were used as the part of the scripts defining the structural models for optimization performed by Karamba 3D. For that reason, grid lines were changed into beams, whereas grid vertices were changed into structural nodes. The assumed boundary conditions regarding the means of support, as well as both joints and material properties, were specified too. The structure was assumed to consist of round steel pipes. The structural nodes of the grid were assumed to be rigid, while the branches' joins with the grid as pinned joins. For a roof covering, polycarbonate plastic panels with a thickness of 10 mm were chosen.

The optimization was performed by Octopus which is the Grasshopper's plug-in for applying evolutionary principles to parametric design and problem solving by multi-objective optimization. Octopus as an evolutionary simulator can approach optimal solution sets through iterative tests and constant self-adaptation. It possesses the ability to cross-reference multiple parameters simultaneously. However, it requires multiple objectives to be input.

The goal of the performed optimization was to determine the best structure in terms of bar grid topology, the location of supports, and the locations of branches' nodes. However, the optimization objectives were as follows:


Structural constrains resulting from general structural principles presented in [31–36] were as follows:


Established variables:


Variables for optimization:


During simulations it was assumed that the structures were composed of round tubes with cross-sections, as expressed in Table 1.


**Table 1.** Division of the structures' bars and their cross sections.

Moreover, it was assumed that each structure is loaded by its self-weight, as well as environmental loads from snow and wind. The wind load was applied locally to the grid structure whereas snow wind was applied globally. These loads were calculated in the form of pressure coefficients acting over the surface of the roof assuming the structure's location in Rzeszow, Poland [34,36]. Several combinations of loads have been considered, including asymmetric ones, which, in the case of canopy roofs, can be crucial when shaping the structures. However, in this case, the worst case scenario was

achieved for the combination when the drifted snow load is a main load and the wind load is an associated load acting from above on the structure.

Due to the symmetry of each roof structure and its shape, some simplifications were proposed; that is, the snow load could be calculated similarly, as in the case of a butterfly roof, whereas roof inclination angle was determined according to Figure 6.

**Figure 6.** Determination of the roof inclination angle.
