# Field-Based Optimal Placement of Antennas for Body-Worn Wireless Sensors
## 2 Methods
### Optimization Algorithm  
Over the last decade, evolutionary algorithms (EAs) have proven to be successful in solving optimization problems in the area of computational electromagnetism, see e.g., [22; 23; 24; 25; 26]. In fact, EAs do not require the objective function gradient to identify the search direction, and are global-optimum oriented in the case of non-convex objectives. Moreover, they are able to find a solution in a short time when the optimization problem is characterized by several constraints or exhibits discontinuities, while classical methods based on gradient information cannot be applied. Often, EAs are inspired by the models of natural phenomena: For instance, \"island\" paradigms mimic the phenomenon of populations evolving without exchange with the external environment [27]. In turn, the biogeography-based optimization (BBO) algorithm [28], which is inspired by the geographical distribution of species in an ecosystem, has been applied in the antenna design [29; 30]. Particle swarm optimization, which imitates the behavior of a swarm of bees during their food-searching activity, was introduced in the antenna community [31] and stimulated numerous contributions, both under single-objective and under multi-objective context [32]. More recently, wind-driven optimization, inspired by the wind motion, which takes place in the high atmosphere to compensate the pressure imbalance, has been proposed and applied in the antenna array design [33]. Numerous techniques have been considered to reduce optimization time, such as more cost-effective electromagnetic codes (e.g., if applicable, the boundary element method instead of the finite difference time domain method), simplified simulation models (e.g., homogenous body phantoms instead of multi-tissue), massively parallel computing (e.g., using graphics card) or surrogate modelling which may require fewer evaluations of the objective function [34; 35]. The approach taken in this paper, which adopts a simplified, homogenous model of the human body, turned out to sufficiently reduce the optimization time combined with an EA EStra [26] lowest order (_i.e._, a single parent generates a single offspring) algorithm. With the direct approach involving Remcom XFdtd and a heterogeneous human body model the computation time turned out to be excessive, considering a great number of design variants taken into account. In this paper EStra [27] was used as an evolutionary algorithm of the lowest order (_i.e._, a single parent generates a single offspring). In EStra each out of \(n_{\circ}\) design variables is represented by two pieces of information, _i.e._, mean value \(m\), and standard deviation \(d\). The search in the \(n_{\circ}\)-dimensional design space begins in a region of radius \(d_{0}\) (initial standard deviation) centered at the starting point \(m_{0}\) (initial mean value); \(m_{0}\) is externally provided, while \(d_{0}\) is internally calculated based on the bounds boxing the variation of the design variables. After initialization, the optimization loop is based on three major steps as follows:  
* Setting \(\mathbf{m}=\mathbf{m}_{0}\) and \(\mathbf{d}=\mathbf{d}_{0}\) at the initial iteration, the generation of the design vector \(\mathbf{x}=\mathbf{m}+\mathbf{d}\mathbf{u}\) takes place, resorting to a stochastic sample \(u\in(0,1)\); generally, \(u\) is a normally distributed sample. Vector \(\mathbf{x}\) is accepted if it fulfils bounds and constraints; otherwise, a new design vector is generated until the corresponding design point falls inside the feasible region. * The associated objective function f(\(\mathbf{x}\)) is then evaluated, and the test f(\(\mathbf{x}\)) \(<\) f(\(\mathbf{m}\)) is performed; if the test is successful, \(\mathbf{m}\) is replaced by \(\mathbf{x}\) (_mutation_ operator), otherwise \(\mathbf{m}\) is conserved.