## 8 Application FVIM to Solve Structural Engineering Design Problems
### Experimental Setup
#### 8.1.3 Cantilever Beam Design Problem  
The Cantilever Beam Design Problem constitutes a pivotal aspect of structural engineering, focusing on the optimal design of a beam fixed at one end, known as a cantilever [15]. Engineers aim to determine the most efficient dimensions and material properties for the beam to withstand specified loads while adhering to various constraints. Key considerations include minimizing deflection, stress, and material usage while maximizing strength and stability. The design variables typically encompass beam length, cross-sectional dimensions, material properties, and support conditions. Engineers leverage analytical methods, empirical equations, and computational simulations to address this optimization problem effectively. Finite element analysis and optimization algorithms facilitate the exploration of the design space, enabling engineers to identify the most suitable beam configuration that balances performance, safety, and cost-effectiveness. Through the Cantilever Beam Design Problem, engineers advance their understanding of structural behavior and develop strategies to create robust and efficient beam designs for a wide range of applications, from bridges and buildings to mechanical components and aerospace structures. The Cantilever Beam Design Problem involves a structure composed of five hollow square cross-sections, as illustrated in Figure 6.  
The mathematical formulation of the problem is as follows:  
Let \(LB=[lb_{1},lb_{2},lb_{3},lb_{4},lb_{5}]\) and \(UB=[ub_{1},ub_{2},ub_{3},ub_{4},ub_{5}]\) denote the lower and upper bounds of the design variables, respectively. Here, \(lb_{i}\) and \(ub_{i}\) represent the lower and upper bounds of the \(i\)th design variable. _Vio_ denotes any violation conditions, and \(Obj\) represents the objective function to be minimized. The global minimum achieved is denoted by \(GloMin\). The objective function, denoted as \(z\), is defined as:  
\[z=0.0624\cdot(x_{1}+x_{2}+x_{3}+x_{4}+x_{5})\]  
where \(x_{1}\), \(x_{2}\), \(x_{3}\), \(x_{4}\), and \(x_{5}\) are the design variables representing certain characteristics of the cantilever beam. Constraints \(g\) are defined as follows:  
\[g_{1}(x)=\frac{61}{x_{1}^{3}}+\frac{37}{x_{2}^{3}}+\frac{19}{x_{3}^{3}}+\frac {7}{x_{4}^{3}}+\frac{1}{x_{5}^{3}}-1\geq 0\]  
Figure 6: Cantilever beam design problem. where \(g_{i}(x)\) represents the \(i\)th constraint function, and \(x_{1}\), \(x_{2}\), \(x_{3}\), \(x_{4}\), and \(x_{5}\) are the design variables. The lower and upper bounds for the design variables are defined by \(LB\) and \(UB\), respectively, and any other conditions such as violation conditions are defined by \(Vio\). As can be seen in Table 20, in the evaluation of optimization techniques for the Cantilever Beam Design Problem, the FVIMDE algorithm has demonstrated a highly effective performance. The statistical measurements for FVIMDE reveal a mean objective function value of 1.340, with an exceedingly narrow standard deviation of \(1.765\times 10^{-7}\), indicating an extraordinary level of precision in FVIMDE's outcomes across various computational iterations. This precision is complemented by the consistency of the maximum value, which precisely matches the mean and best values, denoting a good stability in reaching the optimal solution. The design variables for the minimum value solution, denoted by X1 through X5, have been optimized to 6.017, 5.306, 4.498, 3.501, and 2.152, respectively. These values represent critical dimensions or properties in the cantilever beam design that adhere to the constraints and objectives set forth in the problem. When comparing with the other algorithms, FVIMDE's performance is not only consistent but also indicates a significant margin of superiority in locating the minimum value solution. For instance, algorithms like the GWO and WOA present higher standard deviations and maximum values, which suggest a less consistent approach towards optimization and potential difficulty in reliably identifying the global optimum.