Nowadays, the optimization design is a commonly used procedure applied to many fields of physics and engineering. Depending on the application, the optimization process may improve the performance of the device or the system in many physical aspects of its operation, like mechanics, thermodynamics and electromagnetics . Multiphysics approach requires utilization of complex and computationally expensive simulation software. Another case that is considered in automated optimal design, is one physical domain optimization covering e.g. the electromagnetic properties of the device . Minimizing an objective function, which depends simultaneously on a number of design variables subject to constraints, is a vital problem of many aspects of electromagnetic design in search for innovative or improved solutions. It is the task of e.g. engineers designing the sensors used for detecting high energy neutrino from the outer space  as well as the Micro-Electro-Mechanical Systems designers . This is also an important problem in the design of the wearable antennas that are operating in complex electromagnetic environment such as vicinity of the human body.
The evaluation of the objective function in electromagnetic problems requires the use of field simulations, which may be prohibitively time consuming. Due to the limited computational budget, the algorithms that do not require the objective-function derivatives are the first choice of the designer. In this group of algorithms (so called zero-order) there are local-search oriented (e.g. Nelder-Mead or Powell) and global search oriented (e.g. evolution strategy based). Depending on the application and on the choice of the starting point they differ with respect to computational effectiveness and also with the quality of the solution. Therefore, the appropriate selection of the improvement algorithm is important for the designers.
The overall computational effectiveness of the multi-iterative optimization algorithm depends strongly on the time needed for electromagnetic simulation of the component or device under optimization, which is multiplied by the number of iterations needed for finding the final solution at convergence. For objects, which are significantly larger than the length of an electromagnetic wave the simulation time with numerical methods is significantly larger than the time needed for simulating small elements. This is the case of a wearable antenna that should be simulated jointly with the model of human body that is located in the proximity of the antenna. For the antennas operating in the microwave band the corresponding wave length does not exceed several decimeters. In this case the body model used for simulations has to be discretized with millimeter resolution, which increases the size of the numerical problem to be solved resulting in high memory requirement and the computation time. In this paper we show how the computational effectiveness of the wearable antenna design optimization can be improved by the exploitation of simplified numerical model of human body.
2 Wearable antenna design
The wideband vee antenna, which is considered as the case study, is presented in Figure 1. It consists of a thin-layer type metallic radiator, which is placed on a flexible dielectric material (substrate). This is symmetrical antenna and its feeding points are located on the internal edges of the radiator (see Figure 1). The geometry of the radiator is described by means of 6 geometrical parameters that are indicated in Figure 2.
The following parameters control the geometry of the antenna, namely: L – length of the arm, D – smallest distance between the arms, W – width of the arms, R1 – arm curvature radius, R2 – circle radius that ends the antenna arm, A – the displacement of the last linear section of antenna arm. The dimensions of the base material were fixed to B = 120 mm, C = 100 mm and H = 0.1 mm. The effective dielectric constant of base material was assumed to be εr = 1.7 and the dielectric loss tg(δ) = 0.001. This corresponds to DuPont Pyralux® material, which is thin film polymer covered with copper on one side, that can be successfully used for wearable applications thanks to its flexibility and light weight.
The antenna is designed to operate in the proximity of the human body. In the case of wearable antenna, we have to model the antenna impedance detuning effect caused by small antenna movements, which makes the human body to antenna distance vary. It is usually caused by the relative movements of clothes that carry the antenna on a human body.
To achieve the expected improvement of the design, a large number of electromagnetic simulations may be required. It follows from the necessity of calculating the objective function for several antenna-body distance values for each set of geometric parameters generated by the optimization algorithm. Accordingly, to help the computational cost-effectiveness of the optimization procedure, instead of a full size heterogeneous human body model, the simplified numerical model of human body, which is presented in Figure 3, was used [5, 6]. It consists of two coaxially located cylinders of the height equal to H = 300 mm. The inner cylinder has the radius Ri = 102 mm and is made of material with relative dielectric constant εI = 42.94 and conductivity σi = 2.03 S/m. The outer cylinder has following parameters: Ro = 271 mm, ∊o = 3.35, σo = 0.36 S/m. Thanks to its reduced size and lower complexity, the simulation time was reduced from c.a. 10 minutes for a full-size human body model to 1 minute in simplified model (on the computer that utilizes Nvidia Tesla C2070 GPU card). The antenna was placed at different distances towards body model, which was controlled with x parameter, defined in Figure 4.
3 Experimental procedures: Antenna optimization - analysis and sythesis
The antenna, which was the subject of optimization, can be designed to operate either in free space or in proximity of the human body. In Figure 5 the impedance matching of antenna, that was designed to operate in free space, is presented. It is expected to cover the frequency range from 2.4 GHz to 2.5 GHz with the maximum value for VSWR smaller than 1.3. This can be considered a wide bandwidth of impedance matching, which makes it less sensitive to impedance detuning. In the case of the antenna designed for the free space condition, the antenna impedance matching changes significantly, while placed very close to the human body. In Figure 5 the VSWR, obtained for free space and on-body case, is presented.
The optimization process aimed at finding a set of 6 antenna’s geometrical parameters for which the antenna would obtain the best impedance matching to 50 Ω (lowest value of VSWR) in the frequency range of 2.4 GHz – 2.5 GHz. To calculate the input impedance of the antenna the Remcom XFdtd code that utilizes finite difference time domain method was used . The simulations were controlled by the script that was launched in XFdtd program. For the set of geometrical parameters given by the optimization procedure, the script generated the model of the antenna and launched the simulations of its input impedance vs. frequency. Based on the antenna impedance the objective function was calculated as the maximum value of voltage standing wave ratio - VSWR - within the frequency range of interest. Additionally, 3 different values of antenna distance to the body model (parameter x in Figure 4) equal to 2, 6 and 8 mm were considered and the greatest value of VSWR was taken as the objective function value. The design problem formulation can be cast as follows: given an initial solution (antenna prototype), find the minimizer with respect to design vector g = (L,D,W,R1,R2,A) of the following objective function Ψ (1) should be found:
Ωx – the set of admissible x movements of antenna
B – the ISM f frequency band: 2.4 – 2.5 GHz
Therefore, (1) originates a double min-max problem: in principle, the solution to such class of problems might be non-smooth; therefore, derivative-free optimization algorithms, like evolutionary ones, is recommended. Coherently, the EStra optimization algorithm – an evolution strategy – was used. A version of the lowest order (i.e. a single parent generates a single offspring), which makes the search cost-effective, was chosen. A detailed analysis of the algorithm can be found e.g. in Ref. , while applications in electromagnetic simulations are presented e.g. in [8, 9].
The minimum value of objective function equal to fmin = 1.95 (that corresponds to the maximum value of VSWR) was achieved in 53 iterations. The values of the parameters found by the algorithm had following: L = 74.5 mm, D = 2.6 mm, W = 2.5 mm, R1 = 30.9 mm, R2 = 23.7 mm, A = 6.4 mm.
The antenna impedance matching simulations for the antenna geometry achieved after the optimization process are presented in Figure 6: for antenna distance to body equal to 2 mm – position 1, and 8 mm – position 3.
4.1 Comparative assesment of optimization results
The optimization process of the antenna was subsequently executed with Nelder-Mead simplex direct search algorithm, which was implemented in Matlab function “fminsearch”. This algorithm is very popular for local optimum search and was successfully applied for antenna design improvement [10, 11]. The initial set of design variables was the same as for the case of optimization procedure that used EStra algorithm. The minimum value of objective function found by Nelder-Mead algorithm was equal to fmin = 1.85 (that corresponds to the maximum value of VSWR) was achieved in 176 iterations. The values of the parameters found by the algorithm were: L = 69.3 mm, D = 1.54 mm, W = 7 mm, R1 =48.9 mm, R2 = 24 mm, A = 6 mm.
For the sake of another comparison, the optimization process of the antenna was subsequently executed with Powell conjugate-direction algorithm. Also, this algorithm is very popular for local optimum search and it can be compared with Nelder-Mead because both of them do not use derivative to find the minimum of objective function and therefore they are methodologically equivalent. The initial set of design variables was the same as for the case of optimization procedure that used EStra and Nelder-Mead algorithm. Also the search tolerance was the same for all three algorithms used, and equal to 0.001. The minimum value of objective function found by Powell algorithm was equal to fmin = 1.98 (that corresponds to the maximum value of VSWR) was achieved in 150 calls to the objective function (electromagnetic simulations with XFdtd). The values of the parameters found by the algorithm were: L = 70.1 mm, D = 2.6 mm, W = 5.8 mm, R1 = 38.4 mm, R2 = 22.5 mm, A = 5.4 mm. At the first glance, the computational budget required by Powell algorithm is similar to N-M algorithm, but the final value of objective function is greater than both obtained with EStra and N-M algorithms.
The comparative analysis of three sets of the results assessed the validity of the solution obtained with EStra with lowest computational afford. The impedance matching of antennas optimized with EStra and Nelder-Mead algorithm are presented in Figure 7 for the on body position no.1 (parameter x = 2 mm).
4.2 Assessment of the optimal design
The performance of the antenna that was optimized with EStra algorithm and simplified cylindrical model of the human body was assessed with computer simulations that were performed with the Hershey model as the reference. It is the full scale heterogeneous model available in Remcom XFdtd program . It maps the geometry of adult male and represents the internal structure of the body with 39 different tissues with dielectric properties that correspond to the human body . In Figure 8 the numerical model consisting of the antenna placed towards heterogeneous human body model is presented. The antenna orientation is preserved from the simplified model, where the central axis of cylinders is z axis of the human body model. The distance of antenna to the body surface (x parameter) varied from 2 mm to 20 mm. The computer simulations were carried out to simulate the antenna radiation pattern and impedance matching for various x parameter values.
The antenna impedance matching to 50 Ω is presented in Figure 9. The antenna exhibits good impedance matching to 50 Ω in frequency range of 2.4 GHz up to 2.5 GHz for all considered distances of the antenna to the body, from 2 mm to 20 mm. The VSWR value for the considered ranges of antenna distances is always below 1.75, which is less than the minimum value of objective function. In Figure 10 the gain of the antenna in the horizontal plane (gain versus ϕ angle defined in Figure 8) is presented. The plot is normalized to the maximum value of gain equal to Gmax = 7 dBi. The maximum gain of the antenna changes depending on the antenna to the body distance, from Gmax = 4.4 in x = 2 mm position to Gmax = 7 dBi in x = 20 mm position.
The simulation time for one case of antenna placement towards the human body was approximately 11 minutes on the computer that utilizes Nvidia Tesla C2070 GPU card. The GPU memory used was 1.5 GB.
Design of wearable antennas, which may change their position with respect to the human body during normal usage, is in general computationally costly, because the antenna performance has to be simulated for different distances from the body. Moreover, the number of design variables for the antenna type under consideration and the interdependence between them makes a trial-and-error design method ineffective. This process can be significantly improved with an automated optimization algorithm. This approach was investigated in the presented study.
The optimization procedure presented above uses full wave simulator based on finite difference time domain method to obtain the objective function value based on antenna impedance matching calculated for several values of the antenna-to-body distance. We use a simplified model of the human body to simulate its influence on the antenna impedance. Thanks to the relatively simple structure of the body model and consequently a reduced computational cost, it was possible to use it in multiple simulations in our multi-iterative optimization procedure. The time needed for one simulation was reduced from 11 minutes, that were required in the case of full-scale heterogeneous model, to 1 minute in the case of the simplified cylindrical model.
EStra, which is an evolution strategy based algorithm, proved to be effective for our design problem and allowed to improve antenna design with respect to its impedance matching. The objective function definition that covered both: antenna bandwidth and different antenna distances towards the body, is relatively simple to be derived in optimization process and at the same time is effective for design improvement. To show the numerical effectiveness of EStra algorithm it was compared to the popular Nelder-Mead algorithm. The evolution-strategy based algorithm needed only 53 calls to the full wave simulator, while the simplex algorithm needed 176 calls to the simulator with comparable convergence accuracy. The solutions of the design problem found by both algorithms had very similar performance (see Figure 7). The minimum value of objective function (that corresponds to the maximum value of VSWR over the assumed frequency band and all antenna-to-body distance values) was equal to fmin = 1.95 in the solution given by EStra, and fmin = 1.85 in the solution given by Nelder-Mead algorithm. With approximately 3 times less calls to the simulator, EStra algorithm has shown its computational effectiveness in our antenna design study.
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