Equations 1 through 4 can be also written in integral forms. These four partial differential equations are discretized in a 3D rectangular grid, using a central-difference approximation of two consecutive values for the field components in both space and time. Due to the nature of electric and magnetic fields e.
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In the discretization process, electric and magnetic field components are assigned to the edges of complementary interlocking cubical meshes. Thus, half-space indices are introduced for different field components. Besides, half time steps are introduced to perform the finite difference computation of E based on H and vice versa.
Spatial and time steps are respectively represented by the lower indices i, j, k and the upper index n. Some details of the simulation process are explained in the following. The most advantageous characteristic of the FDTD method is its simplicity. However, there are several limitations to the numerical implementation of the FDTD technique through the above-mentioned difference equations. These limitations have been and are subject of research. One fundamental restriction requires the longest side of the grid cell to be much shorter than the shortest wavelength of the wave within the cell.
This constraint is imposed by both sampling limitation and grid dispersion errors Kosmas, The second restriction develops from the scale and geometry of the problem.
The method uses a uniform grid to model small antennae along with a large soil medium with or without contrasting objects. Accordingly, the geometry imposes challenging limitations, especially in computation cost. For example, in the case of a DNAPL contaminated site, dimensions of the background in horizontal directions may be of the order of tens of meters by tens of meters in the XY-plane, while the antenna thickness may be of the order of millimeters to centimeters.
This requires solving a large uniform grid with a small grid-cell size in the X and Y directions. One way to solve this problem is by using a non-uniform grid, which adds more difficulties to the task of satisfying stability conditions e. The third limitation is the time-step restriction required to satisfy the Courant condition. The Courant condition restricts time increments to a range in which waves do not travel too far in each time increment, or:.
In a 3D case, Equation 9 will be written in the following form:. Different cases have demonstrated that using smaller values of t does not necessarily improve the results. However, smaller values for the Courant number may sometimes yield satisfactory results Kosmas, Soil is a complex medium to model because of its heterogeneous, lossy, and dispersive nature.
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Accurate computation of the behavior of soil over a wideband of frequency requires either several individual frequency domain calculations or a robust deconvolution of E t from D t in the time domain Rappaport et al. Then, the results are inverse-Fourier-transformed. Weedon and Rappaport and Rappaport and Winton simplified the problem by modeling the conductivity as a simple rational function of Z-transform. The real part of conductivity corresponds to measured conductivity, and the imaginary part corresponds to real dielectric permittivity.
As in simulation of any diffusion application e. To model the infinity of the flow or wave propagation through the required infinite media by the finite number of grid cells, appropriate boundary conditions are required. The lattice termination absorbing boundary condition used in the FDTD code was based on the second-order one-way wave propagation equation of Mur and adjusted for lossy soils by Talbot and Rappaport using the rational function approximation of Equation There are some factors that categorize the excitation, such as the source geometry, excitation signal type, and hard or soft source of propagation.
Hard and Soft Sources: The code uses soft sources for all different antenna types.
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A hard source specifies the total field at the excitation point. These types of sources are usually avoided as they cause undesirable reflections at the source point.
Alternatively, a soft source specifies the additional field supplied above the existing background field at the source point. The soft source can be specified at points on a radiating aperture, while hard sources are used for current sources flowing along metal structures. In 2D cases, soft sources propagate well, but extensive testing of 3D FDTD cases has shown difficulties involved in propagation from a soft source unless the physical body of the antennae is modeled. This paper uses soft sources for both monopole and dipole antenna cases of the following sections.
Excitation Signal: A cosine-modulated Gaussian signal Equation 13 was used to excite both antennae because of its simplicity and frequency content. Ideally, a Gaussian pulse has infinite duration.
However, in numerical implementation it ought to be truncated. Thus, W is chosen as the duration of the pulse. Parameters t 0 and W can be chosen, but not absolutely arbitrarily. In this chapter, W was chosen based on the bandwidth used in a scaled experiment by the authors Farid et al. Inappropriate choices of these parameters may introduce instability and noise into the FDTD computation. Starting and truncation of the Gaussian signal should be sufficiently smooth i.
Large spectrum energy of the incident field may cause instability and noise, which in turn, results in computation corruption. This is true for any type of pulse.
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To model the DNAPL detection and evaluate its feasibility, two different antenna types were modeled, a monopole and a dipole. Some fundamental parameters and facts about these simulations are explained in the following. The monopole antenna is simply a coaxial cable, schematically shown in Fig. Different types of antennae have different radiation patterns. The radiation pattern is a visual way of representing how an antenna distributes energy into the surrounding space.
Different methods parameters can be used to demonstrate the wave propagation through space. One popular method is to use power as the key parameter. This method visualizes how and in which direction power is concentrated by the antenna to propagate away. This is possible by drawing contours of equal power. Other parameters, such as field magnitude or phase, can be used for visualization purposes. Contours of these parameters are drawn to visualize radiation patterns. A monopole antenna radiates waves in all different directions.
A dipole antenna consists of two straight metallic or dielectric parts connected at the center to a feed line. A typical dipole antenna is shown in Fig. This antenna type constitutes the main RF radiating and receiving element in various sophisticated types. Dipoles are inherently balanced antennae due to bilateral symmetry. As mentioned before, two different antennae are simulated in the code to model the problem and evaluate the feasibility of the use of the CWR method to detect dielectric objects in soil.
This value of moisture content is selected to model a soil medium similar to the existing fully saturated sandy soil in a pilot-scale facility constructed at Northeastern University referred to as SoilBED for subsurface sensing and imaging experimentation. Hydrological modeling of DNAPLs and the dielectric properties converted through a generic petrophysical relationship in the simulation is desired but does not fit in the extent of this chapter. Details of the two above-mentioned simulated cases follow. In the first case, a 5 mm-thick monopole antenna is modeled within a fully saturated sandy soil background.
The size of the medium under study was selected to satisfy limitations of the FDTD code as well as the experiment.
Table 2 a summarizes details about the geometry and grid size of the soil medium. The simulation is driven by a cosine modulated Gaussian time pulse at a reasonably high frequency 1. To relate the results to the field site, the model can be scaled up in size while scaling down the frequency. The geometry details of the monopole transmitting antenna modeled in this case are tabulated in Table 2 c. The drive signal excites the top of the simulated coaxial cable feeding the monopole antenna in a conventional radial field pattern. The electric field components on all grid points of different cross-sectional and depth slices of the medium were computed and then visualized using MATLAB.
First, the background medium was analyzed.
Then, a rectangular acrylic plate as a representative of a DNAPL pool was modeled within the soil medium. Table 2. Geometry details of the simulated monopole antenna. The dielectric constant and effective electrical conductivity of the extended dielectric of the antenna are respectively assumed to be 2. To evaluate the wave propagation, the following observation slices were selected.
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