## 4 Results and Discussion
### Inverse Problem of Laminate Configuration  
The inverse problem of deciding the laminate configuration to target the lamination parameters for three sets was investigated:  
1. \([V_{1}^{A},V_{2}^{A},V_{3}^{A},V_{4}^{A}]=[-0.45,0.35,-0.55,0.3]\) with PR of \(-0.356\). 2. \([V_{1}^{A},V_{2}^{A},V_{3}^{A},V_{4}^{A}]=[0,0,0,0]\) with PR of \(0.296\). 3. \([V_{1}^{A},V_{2}^{A},V_{3}^{A},V_{4}^{A}]=[-0.55,-0.35,0.05,-0.3]\) with PR of \(1.389\). The relationship between the lamination parameters and the stacking sequence is not always unique; there is no closed-form solution to convert a point in the lamination parameters' space into a stacking sequence. Further still, when prescribing the number of layers in the laminate, certain points in the lamination parameters' space do not have a corresponding layup, and the closest alternative must be found. In previous work, the conversion process was formulated as an optimization problem. The objective function that was used for this optimization was the least rectangular distance of the lamination parameters to the desired optimum point:  
\[ \begin{split}\text{min}f(\theta)&=\sum\limits_{i=1 }^{4}\left[\left(V_{i}^{A}-V_{iopt}^{A}\right)^{2}+\left(V_{i}^{D}-V_{iopt}^{ D}\right)^{2}\right]\\ s.t.&\theta\in\left\{\text{from}-90^{\text{o}}\text{ }90^{\text{o}}\text{ with an interval of }5\text{ degrees}\right\}\end{split}  \tag{28}\]  
where \(opt\) denotes the given optimum lamination parameters. Due to its ability to deal with the continuous global optimization problem with a non-linear objective function, particle swarm optimization (PSO) is an evolutionary global  
Figure 9: Buckling loads with Poisson’s ratio for various in-plane translational restraint coefficients. algorithm that has become more and more popular. PSO was first proposed by [PERSON] and [PERSON] [26; 27]. It is observed that a swarm of birds or insects searches for food in a very typical manner. If one member of the swarm finds a desirable path to go, the rest of the particles will follow quickly. Each particle searches for the best in its locality with learning from its own experience. Additionally, each member learns from the others, typically from the best performer among them. PSO has been successfully applied to some engineering and structural problems. The basic steps in the PSO algorithm are as follows:  
Step 1 Initialize the swarm with random position values and random initial velocities. Step 2 Determine the velocity vector for each particle in the swarm using the knowledge of the best position obtained by each particle and the swarm as a whole, and also the previous position of each particle in the swarm. Step 3 Modify the current position of each particle using the velocity vector and the previous position of each particle. Step 4 Repeat from step 2 until the stop criterion is achieved. The velocity vector of each particle is calculated as follows:  
\[v_{k}^{i}=wv_{k-1}^{i}+c_{1}r_{1}(p^{i}-x_{k-1}^{i})+c_{2}r_{2}(p_{k-1}^{g}-x_{ k-1}^{i}) \tag{29}\]  
where the superscript \(i\) denotes the particle and the subscript \(k\) denotes the iteration number; \(v\) denotes the velocity and \(x\) denotes the position; \(r_{1}\) and \(r_{2}\) are uniformly distributed random numbers in the interval [0, 1]; \(c_{1}\) and \(c_{2}\) are the acceleration constants; \(w\) is the inertia weight; \(p^{i}\) is the best position attained by the particle \(i\) in the swarm so far; and \(p_{k-1}^{g}\) is the global best position attained by the swarm at iteration \(k-1\).