Task 3.3: Simulation of fractal structures in the frequency domain

• Deliverables: D7 T3.3 Final Task Report
• Outputs: Simulation software to be used in WP4.

The presently available software packages for electromagnetic simulation in the frequency domain will be upgraded to analyze fractal structures, with emphasis in wire, planar and microstrip antennas.

Since most simulations will require the solution of huge full-matrix linear systems of equations, the efficient methods (MLFMM, MLMDA, SMA) already developed by partners UPC and EPFL can be upgraded for the analysis of fractal devices.

The aim of Task 3.3 is to support all the other tasks of the project through computer simulations of antenna parameters. The results of the simulations are shown in the corresponding task final report deliverables.

Task 3.3 was originally scheduled to start at T0+6. However, in the kick-off meeting it was decided that task 3.3 started at T0 because UPC was interested in developing algorithms to reduce the computational cost of pre-fractal antennas simulation in the frequency domain.

The numerical methods used for antenna simulations at different groups are:

UPC:

• Surface Electric Field Integral Equation discretized by Method of Moments with Rao-Wilton and Glisson bi-triangular basis functions and free-space Green’s function.
• Thin-wire Electric Field Integral Equation discretized by Method of Moments with linear basis functions and full-kernel evaluation.

The UPC code, FIESTA (“Fast Integral Equation Solver for scaTTerers and Antennas”) has a very efficient iterative solver with multilevel algorithms to achieve a computation time per iteration proportional to N log² N, where N is the number of unknowns. In order to solve ill-conditioned systems of equations, FIESTA has an in-house developed block-LU decomposition algorithm that allows both the use of huge pre-conditioners in the iterative solver and the direct solution of systems of equations having more than 10,000 unkonws.

EPFL:

• Surface Electric Field Integral Equation discretized by Method of Moments with rooftop and quadrangular basis functions (see Task 3.1 final report, deliverable D5) and multilayer Green’s function with Sommerfeld integral evaluation.

UGR:

• NEC public domain code, with thin-wire Electric Field Integral Equation discretized by Method of Moments and enhanced kernel evaluation.
• ? Multiobjective optimization Genetic Algorithms (GA) have been used to find the optimum values of antenna parameters in task 1.2 (see Task 1.2 final report, deliverable D2).

The two new approaches for the analysis of pre-fractal antennas that have been developed at UPC are:

• Optimisation of iterative solver multilevel algorithms to take advantage of the multilevel structure of pre-fractal geometries defined by an IFS.
• Development of a new formulation for a fast evaluation of the full-kernel in the thin-wire Electric Field Integral Equation, in order to allow its application to highly iterated wire monopoles.

Optimisation of iterative solver multilevel algorithms

The numerical analysis of highly iterated pre-fractal antennas by Method of Moments (MoM) involves many tiny subdomain basis functions, resulting in a very large number of unknowns. When these antennas can be defined by an Iterated Function System (IFS), the geometry has a multilevel structure with many equal subdomains. This property, together with a Multilevel Matrix Decomposition Algorithm (MLMDA) implementation in which the MLMDA blocks are equal to the IFS generating shape (Fig. 27), has been used to reduce the computational cost of the numerical simulation of a Sierpinski based structure.

Figure 27: a) Multilevel decomposition for arbitrary shapes. b) Multilevel decomposition for IFS generated fractals: while boxes do not overlap and cover the whole geometry of the antenna, they can be of the same size and shape as the IFS building blocks. This decomposition produces many pairs of source and field boxes with the same interaction matrices.

It has been shown that the combination of the GMRES and the MLMDA scheme, together with the appropriate choice of the shape of the boxes in the multilevel subdivision, leads to a very efficient solution. Our best implementation produces a reduction by a factor of 20 in the total computation time and a factor of 10 in the total memory, compared with a direct application of MoM (Table III).

Table III: Computational requirements to analyze the Sierpinski antenna with the conventional method of moments (MoM) and an iterative solver compared with the optimized approach developed in this project.

Efficient evaluation of the full-kernel of thin-wire EFIE

The thin wire model in computational electromagnetics is used for problems that can be modeled as a set of electrically thin cylinders, referred to as wire segments. The wire segments are assumed to be sufficiently thin for the following conditions to hold:

1. Only the axial components of the induced surface current and the incident field need to be considered.
2. Both the induced surface current and the incident field are constant along the wire circumference.

Under these conditions, the surface current can be expanded in one-dimensional local basis functions for use in the Method of Moments (MoM). In the MoM, the mutual impedances of all the wire segments must be evaluated. This involves the integration of the basis functions and their divergence over the wire surface, with a kernel given by the appropriate Green's function.

If the wire radius a is small compared to the segment length Δ, then an efficient evaluation of this integral is possible using an approximation of the Green's function called the reduced kernel or thin wire approximation. However, there are some problems of practical interest, like highly iterated fractal antennas, in which the model contains short and thick segments, and therefore the thin-wire approximation cannot be rigorously applied.

On the other hand, the integral with the full kernel is valid for any segment length / radius ratio, but it has no analytical solution and solving it by numerical integration is computationally expensive, particularly for the self-impedance of the wire segments, for which the kernel is singular. Several approaches have been presented in literature for the evaluation of the full kernel. The series approach in the extended kernel of the public domain computer program NEC-2 is limited to Δ/a>2. More recently, series representations have been presented that are valid without restrictions on Δ/a. However, for small Δ/a these series show slow convergence.

Task 3.2 “Formulation of numerical methods for fractal structures” showed that the thin-wire reduced kernel of the Electric Field Integral Equation can be used only in low-iteration pre-fractals. Highly pre-fractal wire antennas must be either modeled as a extrusion strip or analyzed using the full-kernel instead of the reduced kernel in the thin-wire Electric Field Integral Equation.

In this task, a new formulation has been developed in order to achieve a very fast evaluation of the full kernel. The computation time if roughly twice the time required for the evaluation of the reduced kernel and the presented formulation is valid for any ratio of the wire segment length Δ to the wire radius a, and anywhere in space, including on the wire surface. Convergence down to Δ/a~10^-4 has been observed, including the usually difficult case of the self-impedance term.

Table IV shows the radiation resistance of a 5-iteration Koch monopole, 6cm height and (1/π) mm radius. The conventional thin-wire reduced kernel results are clearly wrong. The thin-wire with full kernel approach developed in this task agrees very well with the surface formulation of the EFIE using a extrusion strip that, according to the results of Tasks 3.1 and 3.2, can be taken as a reference. The main advantage of the thin-wire full kernel approach over the 2-D formulation is that computation time is very small, since the 1-D discretization requires only 4096 unknowns, while the 2-D mesh has 22,524 unknowns.

Table IV: Radiation resistance at the resonant frequency of a 5-iteration Koch monopole, 6cm height and (1/π) mm radius.