Toy example¶
Incident power density on the surface of a spherical model of human head¶
Power density distribution over a 3-D spherical, homogeneous human head model, diametrically extended to $18$ cm, is shown in Fig. 1.
Consider a canonical setup where center-fed half-wave dipole, set to operating frequency of $10$ GHz, is located at a distance of $5$ mm from the observed averaging area, $A$. Current distribution along the dipole is governed by the Pocklington integro-differential equation and is solved via Galerkin-Bubnov scheme of the indirect boundary element method by using $51$ wire segments, setting the radius of the dipole to $1/10$ of the length of a single segment. Free-space conditions are assumed and are embed into the Green function. Electric field equations arise from boundary element formalism directly, while magnetic field equations are derived from the Maxwell-Faraday law, details are available elsewhere, e.g., [Poljak 2007]. Additional novelty of the approach is the introduction of automatic differentiation when deriving field values, which has been shown to be far superior over numerical differentiation by means of speed and accuracy [Kapetanovic and Poljak 2021].
Figure 2. shows the $xz$-plane of irradiated spherical model of human head from an antenna point of view. By using a red square, the averaging area is focused. In this case, since the operating frequency is set to $10$ GHZ (< $30$ GHz), the averaging area is equal to $4$ cm$^2$, as defined in ICNIRP guidelines.
Gauss-Legendre quadrature is utilized to compute the points and the weights for the averaging area, $A$. The choice of the order of Gauss quadrature will directly impact the accuracy of the overall computed integral. Since averaging area, $A$, is a spherical in shape, the whole geometry and the integration procedure is transformed from Cartesian ($x, y, z$) to spherical ($r, \theta, \phi$) coordinate system. Integration points and weights are thus scaled accordingly, and the magnitude of the integral variable vector normal to $A$ is defined as follows: $$ dA = \Bigg| \frac{\partial r \vec{r_0}}{\partial \theta} \times \frac{\partial r \vec{r_0}}{\partial \phi} \Bigg| d\theta d\phi= r^2 \sin{\theta} d\theta d\phi $$ Finally, the IPD calculated from power density distribution shown in Fig. 3., is integrated by using $33 \times 33$ points in total for $\theta$ and $\phi$ coordinates, respectively.
