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This paper proposes a hybrid technique for treating electromagnetic problems of scattering and radiation in which the source structure is described as an array of antennas. This strategy is based on the combination of the rigorous method multilevel fast multipole algorithm (MLFMA) and the high frequency technique geometrical theory of diffraction (GTD). Thanks to the use of MLFMA, the source can be discretized into several cubic regions considering each of them as a source point in order to reduce the number of times required to compute the ray tracing when GTD is applied to obtain the scatter field. In this analysis, objects with complex shapes are described by using nonuniform rational B-splines (NURBS) which is a very common way to model geometrical bodies. Numerical results that demonstrate the accuracy and efficiency in terms of CPU time are shown.

Traditionally, the method of moments has been used to solve scattering or radiation problems using integral equations [

On the one hand, several rigorous techniques have been developed. One of the most common approaches is the fast multipole method (FMM) [^{2}) to O(N^{3/2}). Its multilevel implementation, MLFMA [

The main objective of FMM and MLFMA is to increase the computation efficiency. However, an alternative solution to the limitation of MoM is to reduce the number of unknowns replacing the type of subdomains basis functions by a set of macrobasis functions [

However, the improvement achieved by the CBFM is not enough when the electrical size of the structure under analysis is very large since the matrix may become very large and the memory needed to store the reduced matrix can represent a problem as well. Therefore, an iterative method is needed to solve the reduced matrix equation. In [

On the other hand, several research works propose the application of high frequency techniques such as geometrical optics (GO) [

In this work, we propose an alternative to take advantage of the MLFMA with the aim of computing electrically large problems applying the high frequency technique geometrical theory of diffraction (GTD) [

The geometrical description of the cases studied in this paper has been modeled using nonuniform rational B-splines (NURBS) [

This paper is organized as follows. Section

The application of MLFMA entails the compartmentalization of the geometry into several cubic regions. Figure

Region division of a generic geometry in the MLFMA.

This method is implemented in three steps: the aggregation, the translation, and the disaggregation step. In the aggregation step, for the first level of cubic regions, the electrical field of each cube is obtained and associated with its geometrical centre, since this point has been chosen to be the aggregation point. This aggregation term is obtained according to Expression (

When the aggregation step is finished, the translations between cubes which are separated but belong to the same level are carried out. Finally, when all the cubes have received the contributions from the rest of cubes in the same level, these contributions are released to the children cubes by means of shifting and

The computation of the matrix-vector products is achieved following this expression:

The aggregation term is calculated as follows:

Analogously, the disaggregation term can be calculated as

The translation term between points

The main goal of the strategy proposed in this paper is to solve radiation problems in which the source structure is composed of an array of multiple antennas. If we try to solve this kind of problems applying GTD directly, this technique needs too much CPU time to obtain the ray tracing for each antenna of the array for the computation of the electrical field radiated by the whole structure. Therefore, to reduce the computational requirements, a new approach based on the combination of the MLFMA and GTD has been developed. In terms of efficiency, this new method allows us to decrease the number of times in which it is necessary to obtain the ray tracing. However, in terms of accuracy, it is worthwhile to remark that every part of the structure shall be at least at a distance

If the radiation pattern of an array of several antennas is obtained applying MLFMA, it is possible to compute the scatter field radiated by the whole geometry considering the source structure as a set of radiation diagrams. Each radiation diagram corresponds to each cube generated by MLFMA.

In this sense, the first step in the computation of the electrical field is the compartmentalization of the source structure into cubical regions. Given an array of antennas like the one shown in Figure

Discretization of a generic array of antennas.

Once we have computed the electrical field for each source applying a fast ray tracing algorithm [

The number of cubical regions depends on the size of the selected window. A typical value of the region size is

In order to give an idea of the computational time saving achieved with the combination of MLFMA and GTD, the radiation pattern of two representative examples has been computed.

The first test case is the Hispasat satellite whose mock-up with a size of 1.6 × 1.4 × 1.5 m is depicted in Figure

Geometrical model of Hispasat with an array of 3 × 3 conical horns.

The radiation pattern at a frequency of 10 GHz and for the set of directions in the

Radiation pattern of the Hispasat at 10.0 GHz, cut

Some conclusions are drawn after analyzing these results. The accuracy of the results obtained with the new technique is very good since the shape of the graphic is very similar to the results obtained with the other two methods. In the computation of the radiation pattern applying GTD, obtaining the ray tracing for the whole set of conical horns is necessary. However, considering a size window of

The main advantage of the new technique is the reduction in the CPU time, as it can be observed in Table

Comparison of the CPU time for the analysis of the Hispasat test case.

GTD | MoM-MLFMA | MLFMA and GTD | |
---|---|---|---|

CPU time | 36 min | 1 h 16 min | 19 min |

The next test case consists of the realistic model of the Passat car, whose dimensions are 4.71 × 1.77 × 1.45 m, shown in Figure

Geometrical model of the Passat with an array of 7 × 7 vertical dipoles.

Figures

Radiation pattern of the Passat at 10.0 GHz, cut

Radiation pattern of the Passat at 10.0 GHz, cut

In this case, considering a size window of

Comparison of the CPU time for the analysis of the Passat test case.

GTD | MoM-MLFMA | MLFMA and GTD | |
---|---|---|---|

CPU time | 2 h 37 min | 3 h 59 min | 9 min |

In this paper, we present the development of a hybrid technique based on the combination of the MLFMA and GTD to obtain the radiation pattern when an array of several antennas is considered. The main purpose of this technique is to reduce the CPU time required to obtain the ray tracing for each antenna of the array without losing accuracy in the results. In this sense, the aggregation step of the MLFMA reduces the number of source points, which implies a reduction in the number of times that is required to compute the ray tracing. The numerical results obtained applying the new approach show good agreement with the traditional high frequency technique GTD or MoM-MLFMA. In spite of these methods, this simple and efficient technique reduces the CPU time required to solve this kind of problems obtaining a high level of accuracy in the results.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work has been supported, in part, by the Comunidad de Madrid Project S-2009/TIC1485 and by the Spanish Department of Science, Technology Projects TEC 2010-15706 and CONSOLIDER-INGENIO no. CSD-2008-0068.