# Optimal Design of Switchable Wearable Antenna Array for Wireless Sensor Networks
## 3 Results  
The optimization process was performed subject to constraints on the possible positions of the four elementary antennas following from the user's clothes design. The azimuthal coordinates of the antennas \(\psi_{a}\) were limited to the range from \(-60^{\circ}\) to \(-10^{\circ}\) for antenna 1, \(10^{\circ}\) to \(60^{\circ}\) for antenna 2, \(120^{\circ}\) to \(160^{\circ}\) for antenna 3, and \(200^{\circ}\) to \(240^{\circ}\) for antenna 4. The initial positions of antennas were \(\psi_{a}=[-20^{\circ}\), \(20^{\circ}\), \(158^{\circ},202^{\circ}]\). The allowable value of phase-shift \(\alpha_{const}\) was within the range from \(45^{\circ}\) to \(270^{\circ}\) and its initial value was \(90^{\circ}\). In this case, the initial value of objective function \(f\) was equal to 0.84. Prior to the optimization, we made a brute-force exploration of the design space. The simplified numerical model that we applied in our research (Equation (2)) is very effective in terms of numerical burden required for the simulation. Therefore, it was possible to explore the entire design space of five parameters with \(5^{\circ}\) steps that gave c.a. 450,000 points within a few hours. The results of this exploration, once mapped in the objective space, are presented in Figures 10-14 in grey color. Although the exploration points have been randomly generated according to uniform probability density, it can be noted that the corresponding distribution in the objective space is not really uniform, being characterized by a few peninsula-shaped branches. This is because functions \(G_{s}\) and \(G_{i}\) are nonlinear functions of the design variables, and nonlinearity determines a non-uniform distribution of points. The min-max approach presented in Section 2.2.1 resulted in 109 iterations, and the value of the objective function was increased to 1.96. The final positions of the antennas are \(\varphi_{a}=[-44.9^{\circ}\), \(45.7^{\circ}\), \(134.8^{\circ}\), \(225.4^{\circ}]\), and the phase shift value was \(\alpha_{const}=117.2^{\circ}\). The optimization process required 7 min of computation time on a PC computer equipped with Intel i7 processor. Figure 10 presents the history of optimization in the objective function components space as well as the results of a brute-force exploration of the design space. The nadir point distance maximization presented in Section 2.2.2 resulted in 145 iterations, and the value of the objective function was increased to 1.87. The optimization process required 10 min of computation time on a PC computer equipped with Intel i7 processor. Figure 11 presents the history of optimization in the objective function component space. The results of a brute-force search of the design space are also presented. Figure 11: The history of optimization in the space of objective function components for optimization with nadir point distance maximization. Figure 12: The history of optimization in the space of objective function components for optimization with utopia point distance minimization. Figure 10: The history of optimization in the space of objective function components for min-max optimization. The utopia point distance minimization presented in Section 2.2.3 resulted in 71 iterations, and the value of the objective function was increased to 1.62. The optimization process required 5 min of computation time on a PC computer equipped with Intel i7 processor. Figure 12 presents the history of optimization in the space of objective function components. The results of a brute-force search of the design space are also presented. The Pareto-optimization presented in Section 2.2.4 resulted in 56 iterations, and the value of the objective function was increased to 1.76. The optimization process required 3 min of computation time on a PC computer equipped with Intel i7 processor. Figure 13 presents the history of optimization in the space of objective function components. The results of a brute-force search of the design space are also presented.