## 2 Introduction to Performative Structural Design Optimization Method
### Computational Design Using Grasshopper  
The proposed methodology starts from a solid beam geometry (Figure 3) whose dimensions are defined as follows: \(L\) (span-length) = 50 m; \(h_{e}\) (depth) = 10 m; \(b\) (shell cross-section) = 0.35 m.  
Figure 3: Test case—start geometry. The just-described parameters represent the constant dimensions of the starting geometry implemented in Grasshopper [24] (Figure 3). The considered shape is described by means of a set of physical parameters:  
\[\left\{p_{i}\right\}_{i=1,\cdots,N} \tag{1}\]  
including the optimization variables. In this case, the solver can act by emptying the solid beam, in the respect of certain parameters that, for the test case, also represent geometrical constraints and reflect the preliminary creative idea on the basis of which the conceptual design develops. The first emptying function is retrieved starting from the general circumference equation, centered with respect to the \(z\)-axis:  
\[x^{2}+y^{2}+\beta z+c=0 \tag{2}\]  
It is worth noting that, for the bottom profile of the beam, a circumference arc shape, that is a constant curvature outline, is preferred, at the aim to simplify the construction stage, also with reference to a possible prefabrication system, and to reduce costs. The arc described by Equation (2) can be parameterized by imposing its passage through three points (Figure 4), named \(P_{1}\), \(P_{2}\), \(P_{3}\), characterized by the following coordinates:  
\[P_{1}=(-\text{L},\,0)\]  
\[P_{2}=(0,\,h_{m^{\prime}})\]  
\[P_{3^{\prime}}=\left(L,\ \frac{h_{e}}{2}\right)\ \Rightarrow\ \ \frac{h_{e}}{2}\ (const.) \tag{10}\]  
In fact, with the aim to achieve both a bigger number of voids to introduce in the beam and more flexibility in their shape, so as to allow a better distribution of stresses, the ellipse shape is selected, under the hypothesis that all the centers of the ellipses are placed on the curve described by Equation (7). Moreover, in order to simplify the problem from a computational point of view, the condition of symmetry is set. Figure 4: (**a**) Emptying function—the arc of circumference passing through three points; (**b**) Grasshopper development of the emptying function (upper definition in red). Figure 5: Shifted curve (Equation (8)). The ellipse equation is so considered:  
\[\overline{PF_{1}}+\overline{PF_{2}}=2a \tag{11}\]  
that can be rewritten as:  
\[\sqrt{\left(x-x_{1}\right)^{2}+\left(y-y_{1}\right)^{2}}+\sqrt{\left(x-x_{2} \right)^{2}+\left(y-y_{2}\right)^{2}}=2a \tag{12}\]  
and that, in parametric form, becomes:  
\[\left\{ \begin{array}{l}x=a\cos t\\ y=b\sin t\\ 0\leq t<2\pi\end{array} \right. \tag{13}\]  
In the case under examination, the \(x\)-axis is replaced by the curve described in Equation (7) and the \(y\)-axis is replaced by the \(z\)-axis. With reference to the ellipse, it is imposed that: \(a\) = horizontal semi-axis; \(2a\) = horizontal axis; \(b\) = vertical semi-axis; \(2b\) = vertical axis. So, it is possible to recognize the major axis by comparing the terms \(a^{2}\) and \(b^{2}\) as follows:  
\[a^{2}\geq b^{2}\ \Rightarrow\ \overline{PF_{1}}+\overline{PF_{2}}=2a \tag{14}\]  
\[a^{2}<b^{2}\ \Rightarrow\ \overline{PF_{1}}+\overline{PF_{2}}=2b \tag{15}\]  
Within this configuration, the circumference shape is included by setting:  
\[a^{2}=b^{2}\ \Rightarrow\ x^{2}+y^{2}+\alpha x+\beta z+c=0 \tag{16}\]  
In this way, three possible solutions regarding the shape of voids are obtained: ellipses with a major horizontal axis (Equation (14)), ellipses with a major vertical axis (Equation (15)) and circumferences (Equation (16)). The shape of voids belongs to the physical parameters set in Equation (1). Equation (14) is strictly dependent on the position of point \(P_{2}\) (and on the parameter \(h_{m^{\prime}}\) (Equation (3)). As the height of point \(P_{2}\) increases, the ellipse (or circumference) must maintain a size to ensure a sufficient thickness between the perimeters of the void and of the beam, so providing a suitable rigidity and strength to the structural component.