## 4 Methodology for Design and Optimization of Material-Structure Integration  
### Definition of the Optimization Model  
In the car door design process, multiple, often conflicting, objectives arise [27]. For instance, increasing the fiber volume fraction enhances door stiffness but may compromise its weight reduction. To balance various performance indicators of the car door, this paper selects the mass \(M\) of the CFRP car door and the sagging displacement \(d_{\text{s}}\) as the objectives, with the upper torsional displacement \(d_{\text{U}}\), lower torsional displacement \(d_{\text{L}}\), first natural frequency \(f\), Tsai-Wu failure criterion _TWSI_[28], and layup process as constraints. The optimization mathematical model is constructed as follows:  
\[\begin{cases}\text{Find}:&\mathbf{X}\\ \text{Minimize}:&M(\mathbf{X}),\ d_{s}(\mathbf{X})\\ \text{{Subject}\ to}:&d_{\text{U}}(\mathbf{X})<10\ \text{mm}\\ &d_{\text{L}}(\mathbf{X})<7\ \text{mm}\\ &f(\mathbf{X})>35\ \text{Hz}\\ &0\leq TWS1\leq 1\\ &\mathbf{X}=[x_{1},x_{2},x_{3},x_{4}\cdots x_{15}]^{\text{T}}\end{cases} \tag{5}\]  
where design variables \(x_{1}\)-\(x_{3}\) are continuous variables, variable \(x_{1}\) represents the half-side length of the microscale cell (as shown in Figure 2), variable \(x_{2}\) represents the width of the fiber yarn (as shown in Figure 5), and variable \(x_{3}\) represents the thickness of the fiber yarn (also shown in Figure 5). Design variables \(x_{4}\)-\(x_{15}\) are discrete variables, with each design variable offering a choice among four layup angles: \(-45^{\circ},0^{\circ},45^{\circ}\), and \(90^{\circ}\).  
To ensure that the optimization results comply with the layup process constraints for composite materials, this paper proposes the following implementation methods for common process constraints:  
* Layup orientation principle, selecting \(0^{\circ}\), \(90^{\circ}\), and \(\pm 45^{\circ}\) as the layup angles;
* Discrete integer variables 0, 1, 2, and 3 are used to represent layup angles of \(-45^{\circ}\), \(0^{\circ}\), \(45^{\circ}\), and \(90^{\circ}\), respectively, with \(N_{u}\) denoting the angle of the \(u\)th layup. During the initialization and particle updating stages of the Multiple Objective Particle Swarm Optimization (MOPSO) algorithm [29,30], the layup orientation constraint is enforced by modifying the methods for generating and updating particles.
* Balanced symmetry layup principle;  
The balanced symmetry layup requires that the laminate is designed with a symmetrical structure during layup design, effectively preventing warping of the laminate under load. This constraint can be represented by Equation (6):  
\[N_{u}-N_{13-u}=0\ u=1,2,\cdots 6 \tag{6}\]  
In Equation (6), when the layup angle of the \(u\)th layer is between \(0-3\), the layup angle of the (13-\(u\))th layer is also between \(0-3\) and appears paired with it.
* \(\pm 45^{\circ}\) single-ply symmetrical layup principle  
To prevent warping deformation of the laminate under load, when there are \(\pm 45^{\circ}\) layups in the laminate, there must be \(\mp 45^{\circ}\) layups appearing in pairs with them. The implementation of this constraint is shown in Equation (7):  
\[\begin{array}{ll}N_{13-u}=0&N_{u}=2&u=1,2,\cdots 6\\ N_{13-u}=2&N_{u}=0&u=1,2,\cdots 6\end{array} \tag{7}\]  
In Equation (7), when the layup angle of the \(u\)th layer is 2 (\(45^{\circ}\)), the layup angle of the (13-\(u\))th layer is 0 (\(-45^{\circ}\)); similarly, when the layup angle of the \(u\)th layer is 0 (\(-45^{\circ}\)), the layup angle of the (13-\(u\))th layer is 2 (\(45^{\circ}\)).  
* Avoiding the consecutive layup of four layers with the same angle.  
When a composite laminate has too many identical and consecutive layups, there is an increased risk of delamination. To prevent this, generally, no more than four layers of the same layup are allowed in the design of composite materials. This manufacturing constraint is managed by evaluating the design variables generated during the optimization process to avoid such occurrences.