**3. Result**

The simulation was carried out four times: twice for structures with the grid pattern of type a and twice for structures with the grid pattern of type b, presented in Figure 4. The first simulation for both structures was carried out assuming bar lengths within the scope of 1.5–2.0 m, while the second one assumed bar lengths within the scope of 1.0–3.0 m.

Due to the fact that the optimization objectives indicated previously—the minimization of total mass and minimization of deflection—are conflicting, several results of each simulation for both structures with pattern a and b have been chosen. The graph of the Pareto front with the best solutions for the structure of type **a** received during the first simulation is presented in Figure 7. The individuals that are displayed closest to the origin are equally optimal for all three objectives, however, in our case solutions which meet both ULS and SLS were chosen. As was mentioned earlier, ULS and SLS are verified automatically. However, deflection cannot exceed 0.04 m. The generated solutions are characterized by the fact that the greater the mass of the structure, the smaller the deflection. Therefore, several solutions were chosen for which the deflections are close to but not exceeding 0.04 m. This guarantees the minimum mass of the structure.

**Figure 7.** The 2D graph of the Pareto front for the hyperbolic paraboloid canopy structure with pattern **a** (split along the short diagonal—division into eight parts).

The chosen results of the first simulation performed for the structure of type a (grid split along a short diagonal) are given in Table 2. However, the results of the simulation performed for the structure of type b (grid split along a long diagonal) are given in Table 3.


**Table 2.** The results of the simulation assuming bar lengths within the scope of 1.5 m–2.0 m.


It is worth noticing that the achieved bar grids during the first simulation were obtained as a result of the division of the surface into eight parts in both directions. Due to the smallest mass, the structure a1 is the most efficient, Figure A2.

During the second simulation, a maximum bar length of 3 m was allowed and bar grids were obtained as a result of the division of the surface into four parts in both directions. The Pareto front for the structure type b is presented in Figure 8. However, several solutions of the simulation assuming bar lengths within the scope of 1.0–3.0 m are presented in Tables 4 and 5.

**Figure 8.** The 2D graph of the Pareto front for the hyperbolic paraboloid canopy structure with pattern **b** (split along long diagonal—division into four parts).

**Table 4.** The results of the simulation assuming bar lengths within the scope of 1.0–3.0 m.



The analysis of the above results suggests that there was a certain reserve in the amount of deflections in the structure, which created the possibility of reducing bars' cross-sections. Therefore, another simulation was performed assuming bars' cross-sections as in Table 6.

**Table 6.** Division of the structures' bars and their cross sections.


The results of this simulation for structures of type a and type b are given in Tables 7 and 8, respectively. However, the optimum results generated due to performed simulation are presented in Figures A3 and A4.


**Table 8.** The results of the simulation assuming bar lengths within the scope 1.0–3.0 m.


On the basis of the above, the results of the most efficient structure resulted in the structure a1, presented in Figure 9.

**Figure 9.** Perspective view of the optimal structure; the structure of pattern a, received due to surface division into four parts along its edges (structure a3, T7).

### **4. Discussion**

An original algorithmic-aided method of shaping the hyperbolic paraboloid canopy roof structures has been proposed. This method verifies simulation structures' geometries with respect to structural requirements. The performed analyses found several solutions that meet ULS at given boundary conditions and, at the same time, meet SLS, i.e., do not exceed the allowable deflection of 40 mm. Based on the results compiled in the tables T2, T3, T4, T5, T7, T8, it can be concluded that the mass of the considered structure depends on the grid topology. In turn, this topology is dependent on the method of surface division, as well as the number of divisions applied. The curvilinear steel bar structures resulting from the division of the hyperbolic paraboloid into four parts are much lighter than the structures resulting from the division into eight parts. Due to the fact that the weight of the structure significantly affects its cost, these structures are more effective.

As previously mentioned, triangular grids of curvilinear steel bar structures analyzed in the study are obtained by dividing spatial quadrate grids, and the division can take place along the longer diagonals or shorter ones. The research revealed large differences in the masses of the structures depending on the shaping of triangular grids based on quadrangular grids. The analysis of the structures carried out showed that the grid structures obtained by dividing quadrangles along the longer diagonals are much heavier than the grid structures formed when dividing them along shorter diagonals.

The location of the supporting columns is another aspect that has a significant impact on the efficiency of the structure. The simulations showed that the further the columns are moved away from the edge of the covered square, the larger the mass of the structure. Moreover, the research found the optimal branch node positions, and thus the optimal column lengths for each structure.

### **5. Conclusions**

The optimal design of structures is an important direction for the development of research. The presented work is a contribution to the research conducted in the field of design optimization by modern digital tools. The studies have shown that the multi-objective optimization does not give one unambiguous optimal solution, especially when the assumed criteria are contradictory. However, it can pre-estimate solutions and select the most favourable ones. The solutions selected due to multi-objective optimisation should be subjected to further analysis and selection in order to choose the most optimal result. However, this kind of optimization at the initial stage of shaping has a justification when it is difficult to assess the behaviour of the structure and choose the right solution.

Due to its properties, a hyperbolic paraboloid always constitutes an important basic shape for various interesting single or complex architectural forms of different types; therefore, this study should be continued. It is very important to take into account other optimization criteria of the structure, such as the unification of elements and the method of connection, which will be a goal of the author's further research.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
