## 8 Application FVIM to Solve Structural Engineering Design Problems
### Experimental Setup
#### 8.1.1 Tension/Compression Spring Design Problem  
The Tension/Compression Spring Design Problem (See Figure 4)is a pivotal optimization challenge in mechanical engineering, aiming to minimize the spring's weight while adhering to various physical and geometric constraints [64]. This entails optimizing key design variables such as wire diameter, mean coil diameter, and the number of active coils, which influence the spring's mechanical properties. The choice of material also plays a critical role in determining the spring's strength and resilience, with manufacturing considerations impacting the feasibility and cost-effectiveness of the design. Engineers tackle this problem using a blend of analytical methods, empirical formulas, and computational techniques like finite element analysis to simulate the spring's performance under load, and optimization algorithms to efficiently navigate the design space. The ultimate goal is to develop a spring that meets specific load-bearing requirements, durability standards, and application-specific conditions, embodying the integration of theoretical principles, practical constraints, and advanced optimization strategies in engineering design. The primary goal is to minimize the fabrication cost of a spring, governed by three parameters and four constraints. These include wire diameter (\(x_{1}\)), spring coil diameter (\(x_{2}\)), number of active coils (\(x_{3}\)), and constraints related to deflection (\(g_{1}(\mathbf{X})\)), shear stress (\(g_{2}(\mathbf{X})\)), surge frequency (\(g_{3}(\mathbf{X})\)), and outer diameter limit (\(g_{4}(\mathbf{X})\)). The mathematical formulation of the problem is as follows:  
Let \(LB=[lb_{1},lb_{2},lb_{3}]\) and \(UB=[ub_{1},ub_{2},ub_{3}]\) 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:  
\[z=x_{1}^{2}\cdot x_{2}\cdot(x_{3}+2)\]  
where \(x_{1}\), \(x_{2}\), and \(x_{3}\) are the design variables representing certain characteristics of the spring. Figure 4: Tension/compression spring design problem. Constraints \(g\) are defined as follows:  
\[g_{1}(x) =1-\frac{x_{2}^{3}\cdot x_{3}}{71785\cdot x_{1}^{4}}\geq 0\] \[g_{2}(x) =\frac{4\cdot x_{2}^{2}-x_{1}\cdot x_{2}}{12566\cdot(x_{2}\cdot x _{1}^{3}-x_{1}^{4})}+\frac{1}{5108\cdot x_{1}^{2}}-1\geq 0\] \[g_{3}(x) =1-\frac{140.45\cdot x_{1}}{x_{2}^{2}\cdot x_{3}}\geq 0\] \[g_{4}(x) =\frac{x_{1}+x_{2}}{1.5}-1\geq 0\]  
where \(g_{i}(x)\) represents the \(i\)th constraint function, and \(x_{1}\), \(x_{2}\), and \(x_{3}\) 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 18 for the Tension/Compression Spring Design Problem, the FVIMDE optimizer resulted in efficient outcomes. An analysis of the results demonstrates that the mean objective function value obtained by FVIMDE is 1.2665E-02, the most competitive across all optimizers considered in this study. The consistency of the FVIMDE algorithm is validated by an exceedingly low standard deviation of 1.4363E-09, indicative of a high level of result repeatability and algorithmic robustness. This suggests that the variability in the quality of solutions across different executions is negligible, signifying a stable convergence towards the global optimum. Further scrutiny reveals that the differences between the maximum values and the minimum values are small, which solidifies the assertion of FVIMDE's steadfast performance in each independent run. The parameters corresponding to the optimal design dimensions, denoted as X1, X2, and X3, are identified as 5.1686E-02, 3.5663E-01, and 1.1294E+01, respectively.