Material Mapping Algorithm

solution the uniform discretization of the interval]xj,ref, xj+1,ref[. Minimize the objective functionf(x e ) using thesteepest descentline search method [34];

4. If the value off(x e

opt)is smaller than a previously defined value, which for the developed algorithm is by default 1%, the mesh satisfies the necessary requirements for the current value ofNj and the algorithm jumps to step 5. Otherwise, setN_j ←N_j+ 1 and return to step 3, unlessN_j> N_upper, which means that the algorithm has not been able to create a mesh with the desired characteristics. For the latter case, the algorithm chooses thex

e

optsolution that leads to the lowest error, displays a message warning the user about the problems that occurred during the discretization of the current interval and proceeds to step 5;

5. Add the elements of the arrayx e

opt to theX set; Setj←j+ 1and go to step 1 unless all intervals of X_refhave already been discretized. In this case the procedure ends, since theX set has been completely determined.

Although this approach may not lead to a globally optimal solution, which could only be achieved by solving the optimization problem for all elements ofX_ref simultaneously, it will nevertheless be very close to this solution as the lengths of the cells adjacent to the lines ofXrefare superiorly limited byQ Lmin, which, to some extent, decreases the interdependence between the meshing of adjacent intervals.

The mesh generated using this algorithm, for the example of Fig. 7.2, is shown in Fig. 7.3. As can be observed, the mesh lines are aligned with the boundaries of the geometry, where the mesh resolution is close to Dmax = 30. For the remaining regions, the mesh resolution is close toDmin = 10, which proves that the developed spatial discretization algorithm indeed reduces the number of cells required to discretize the structure.

Figure 7.3: Two-dimensional Cartesian grid that results from the non-uniform discretization of the structure of Fig. 7.2 using the spatial discretization algorithm with parameters Dx,min=Dy,min= 10and Dx,max = Dy,max= 30.

Supposing that the computational domain’s size is Nx by Ny by Nz cells, the algorithm performs the material mapping by successively processing Nz two-dimensional cut-planes perpendicular to thezaxis, of size Nx by Ny cells. To illustrate how this is attained, consider the situation depicted in Fig. 7.4(a), which represents a two-dimensional view of the Cartesian grid that results from the discretization, using the algorithm described in the previous section, of a component consisting of three overlapping cylindrical structures, resembling a coaxial cable. The cross section of the component is also represented in Fig. 7.4(a). It is important to note that the resolution of the grid was intentionally chosen to be quite low in order to make the description of the algorithm’s operation clearer.

Figure 7.4: Illustration of the material mapping algorithm’s operation. (a) Two-dimensional view of the component being meshed and the Cartesian grid that resulted from spatial discretization; (b) Operation of the ray-crossing algorithm.

The method used to determine which material should be assigned to each cell is based on a modified version of the ray-crossing algorithm used in [36] and [37]. This algorithm scans the computational domain by “shooting” rays along thexdirection and determining the intersection points, if they exist, with the three- dimensional models. The intersections are performed using the functionalities available in the Java 3D API [5], namely the PickTool and PickInfo classes. The rays are cast from a position outside the computational domain so that they pass through the centers of the cells whose material properties are being determined, as shown in Fig. 7.4(b).

Taking into account that before starting the material mapping stage all mesh cells are set as being made of the same background material, which by default is vacuum, then, if no intersections are detected, as is the case for rayR1 in Fig. 7.4(b), no modifications to the material mapping are required. If, however, the number of intersections with a structureSu,Nu,int, is different from zero, as is the case for raysR2,R3 and R4 of Fig. 7.4(b), the information obtained from the intersections’ x-coordinates is used to create, for each intersected structure,Nint,S_u/2intervalsIS_u,j, having the form

IS_u,j = [xS_u,2j, xS_u,2j+1] , j= 0, ...,Nint,S_u

2 −1 (7.10)

where the assumption that Nint,S_u is an even number derives from the fact that the number of intersections for a closed surface will always be even. For this reason, the algorithm will neglect structures for which this number is odd, displaying a message alerting the user to the occurrence of the problem.

Taking into account that the center coordinates of each cell crossed by the ray can be calculated using (7.2), it is easily concluded that a cell with indexibelongs to structureSu if its center coordinatecx,iis

contained by any of the intervalsIS_u,j. For the case of rayR2 of Fig. 7.4(b), the intersected structure is the cylinder with the biggest radius,S₀. Since the number of intersections is 2, the algorithm only creates a single intervalIS₀,0= [x0,0, x0,1] = [x4, x7], which contains the centers of the cells with indices(i= 4, ...,9;j= 12), which are shown in blue in Fig. 7.4(b). As a result, these cells’ material is set to PEC.

A different situation occurs for rayR3, which intersects twice bothS0andS1, originating the inter- vals IS0,0= [x₂, x₁₀]andIS1,0= [x₃, x₈]. A conflict now arises as the centers of cells(i= 3, ...,10;j= 10) are contained by both intervals. Since this component should model a coaxial cable, it is desirable that the dielectric material ofS₁is assigned to cells(i= 3, ...,10;j= 10). In order to achieve this, a mechanism based on priorities similar to the precedence method used in [38] was devised.

Basically, before starting the mesh generation stage, the user should assign different priorities to overlapping structures having different materials. For the case ofR3,S1should be able to “steal” cells fromS0

and therefore the former should have a higher priority than the latter. The priorities are used by the algorithm in the following manner:

1. Each intervalIS_u,j receives the priority of the corresponding structureSu; 2. The intervals are sorted according to their priority;

3. The algorithm processes the intervals one at a time, in ascending priority order.

For the case of R₃, it can be seen that by applying the previous procedure, IS0,0 is processed first and therefore cells (i= 2, ...,11;j= 10) are set to PEC. Then, interval IS₁,0 is processed and cells (i= 3, ...,10;j= 10)are remapped to the dielectric of structure S₁, which leads to the desired mapping of materials. A situation similar to that ofR3 also occurs for rayR4, with the difference residing in the number of conflicting intervals, which now is 3. It can be solved simply by assigning to S2 a priority that is higher than those ofS₀ andS₁.

In order to demonstrate the usefulness and effectiveness of the priority mechanism, 3D views of the meshed version of the component of Fig. 7.4(a), using different priorities for the structures, are shown in Fig. 7.5. While Fig. 7.5(a) corresponds to the case of the coaxial cable, Fig 7.5(b) represents a cylindrical waveguide, which was achieved by decreasing the priority of structure S2 and by changing the material of structure S1 to vacuum.

Figure 7.5: Illustration of how the assignment of different meshing priorities to the elements of a three- dimensional model can influence the characteristics of the resultant component. (a) Three-dimensional view of the mesh for a coaxial cable; (b) Three-dimensional view of the mesh for a cylindrical waveguide.

Chapter 8

Numerical Experiments and Results Validation

In order to validate the software’s implementation, some numerical experiments were performed to study antenna problems for which experimental and/or simulation results had already been obtained by other authors. In general, the obtained results show a good agreement with experimental and simulated results, which indicates that the software is sufficiently accurate to be used for engineering purposes.

8.1 Dipole Interaction

This experiment, published by Luebbers and Kunz [39], models the interaction between two dipole antennas. The quantities of interest are self- and mutual antenna admittances, which are quite sensitive to the geometry of the wires (including the wire radius) and the spacing between the two antennas. The problem’s geometry is shown in Fig. 8.1.

Figure 8.1: FDTD problem space for the analysis of coupling be- tween two dipoles.

Figure 8.2: Mesh used to discretize the computational domain.

Figure 8.3: Self-admittance for antenna 1 calculated with the developed software and compared with results obtained with other methods/applications.

The input waveform is a gaussian signal with a -20 dB bandwidth of 1000 MHz, which is fed to the antenna using a 50Ωresistive voltage source that uses a virtual unidimensional transmission line, as described in Sec. 5.3. The computational domain was discretized using an uniform mesh, shown in Fig. 8.2, that has a density of 20 cells per wavelength in the xandy directions and a density of 30 cells per wavelength alongz.

The domain’s size is 28.5 cm x 18 cm x 73 cm and 19 x 12 x 73 cells are used for the discretization along the x,y andz axes respectively. The thin-wire model presented in Chapter 4 was employed to correctly account for the wires’ radius of 0.281 cm and an UPML layer with a thickness of 10 cells was used to absorb the outgoing electromagnetic waves.

Although Luebbers & Kunz only published data forY₁₁ andY₁₂, in this experiment the full admit- tance matrix was calculated in order to verify if the developed software preserved port reciprocity. For this purpose two simulations were performed: one with the first dipole being fed and the second dipole’s terminals short-circuited and another with the latter being fed and the former short-circuited. For both simulations a time-step∆t= 23.04ps was used and the number of simulated time-steps was 1690 for dipole 1 and 1280 for dipole 2, which was enough to allow the total energy to decay about 40 dB relatively to the maximum value.

The results forY₁₁ andY₁₂ are shown in Figs. 8.3 and 8.4 and are compared not only with those published by Luebbers and Kunz, using both FDTD and a MoM code (ESP4)¹, but also with the results obtained with CST Microwave Studio’sR [40] transient solver, a commercial application that has a time- domain solver based on FDTD/FIT [41]. It can be seen that the obtained results are in good agreement with those obtained with the MoM code, which is considered a quasi-analytic solution for problems involving wire antennas.

The results forY₁₂ andY₂₁ are shown in Fig. 8.5. As can be seen, the results are almost a replica of each other, which means that the admittance matrix is symmetrical, as predicted by theory.

1Electromagnetic Surface Patch Version 4

Figure 8.4: Mutual admittance calculated with the developed software and compared with results obtained with other methods/applications.

Figure 8.5: Comparison of the mutual admittances Y12 andY21 calculated with the developed software, for the purpose of port reciprocity analysis.