## 4 Optimization Mathematical Model
### Topology Optimization Methodology  
Topology optimization is a mathematical method used to formulate, and then to determine an optimum distribution of a material within a pre-defined design space. An optimized structural material configuration within a design domain can be obtained by considering its specific boundary conditions and constraints. The structural design fundamental materials are treated as solid, whereas null materials are regarded as void. Hence, the unnecessary structural weight can be eliminated. A commercial topology optimization application, Atair's OptiStruct, is used in this investigation to manage the given design constraints by defining the objective functions for individual load cases. This application was designed to manage the constraints with the aim to ensure that design specifications were satisfied; therefore, in this paper, essential materials of the design are established, and the process to eliminate redundant elements is defined. Minimizing the compliance while maximizing the structural stiffness under specific design constraints can indeed meet the design requirements. As mentioned above, the TO used in this study is combined with Finite Element Analysis (FEA), while the density-based approach is conducted. The design loads and parameters are applied to the wing components in order to deal with the material density distribution and its other mechanical properties (displacement, stresses, etc.). TO is based on the Solid Isotropic Material with Penalization (SIMP) method. The design domain is optimized by minimizing the objective function, which refers to the structural compliance. The objective function variable is the material density, that defines whether a finite element is solid or void. The pseudo-density, \(x_{i}\), of the _ith_ element may take values between 0 and 1, or \(x_{i}\) (\(0\leq x_{i}\leq 1\)), wherein the 0 value represents a material as void, and the 1 value represents the material as solid. The pseudo-density variable can be defined with the following equation:  
\[x_{i}=\frac{\rho_{i}}{\rho_{0}} \tag{4}\]  
where \(\rho_{i}\) is the density of the _ith_ element, \(\rho_{0}\) is the density of the base element, and \(x_{i}\) is the pseudo-density of the _ith_ element. The formulation of the SIMP method defines the relationship between the material pseudo-density variable and its stiffness properties, as follows [36]:  
\[E_{(x_{i})}=E_{solid}(x_{i})^{p} \tag{5}\]  
where \(E_{solid}\) denotes the base material's isotropic property and \(p\) is the exponent of the penalty parameter. Equation (5) can then be expanded the using the young modulus for solid and void regions:  
\[E_{(x_{i})}=E_{void}+x_{i}^{p}(E_{solid}+E_{void})\hskip 28.452756 ptp\geq 1 \tag{6}\]  
where \(E_{void}\) is the low [PERSON]'s modulus assigned to void regions, \(E_{solid}\) is the [PERSON]'s modulus assigned to solid regions, and \(E_{(x_{i})}\) is [PERSON]'s modulus assigned to each element. The penalty factor value, \(p\), should be selected to be large enough (\(p\geq 3\) is generally considered), so that the volume constraint is active when the intermediate densities are penalized. Since the intermediate densities are close to 0, the material is unnecessary and may be therefore considered void. Contrarily, if the intermediate densities are close to 1, the material is critical to the structure's integrity and can be considered as solid [37]. The volume fraction is a constraint to the total volume of the design domain related to the pseudo-density, as represented in the following equation:  
\[V=\sum_{i=1}^{n}x_{i}V_{i} \tag{7}\]  
where \(V\) denotes the total volume and \(V_{i}\) expresses the volume of the _ith_ element. The objective function of the TO is to minimize the structural compliance, that is subjected to the design constraints and can be defined with the following mathematical statement:  
\[