Combining 3D Finite Elements and Padé Approximations

Combining 3D finite elements and Padé approximations

for wide band antennas analysis

Brahim Essakhi, Lionel Pichon

Laboratoire de Génie Electrique de Paris UMRS 8507 CNRS, SUPELEC, UPS, UPMC

Plateau de Moulon, 91192 Gif-sur-Yvette cedex, France,

Abstract— A 3D computationally-efficient numerical model which combines the edge finite element method (FEM) with a Padé rational approximation procedure is presented for solving antennas problems. The technique allows to obtain the solution over a wide frequency band from only a small number of full wave computations. The ability of the model is shown in the case of patch antennas for which the impedance is obtained as a function of frequency. The comparison of the proposed approach with a standard finite element method shows that the computational cost is significantly reduced.

Keywords—finite element method, Padé approximation, patch antennas

I. Intoduction

In many electromagnetic wave propagation problems (antennas, electromagnetic compatibility) it is often necessary to compute the frequency response over a wide frequency band. With a frequency domain finite element analysis the electromagnetic fields the unknowns are solutions of a linear system whose matrix depends on frequency. In a wide frequency band analysis the linear system has to be solved for each frequency of interest. This often leads to a huge computational cost. An alternative approach is to search for an approximation of the solution about a center frequency or over a frequency band. In wave propagation problems the field’s behaviour includes resonance phenomena with sharp peaks: a polynomial approximation cannot give a suitable fit. A substantially better accuracy can be obtained with rational approximations.

Padé-based methods can provide efficient techniques for wide band computation. They were first shown to be very efficient for electromagnetic problems within bounded domains where passive microwave devices such as waveguides and cavities were studied [1-3]. In these almost closed structures the efficiency of the approximation relies on the fact that dominant poles and zeros of the network transfer function can be used to build rational approximations of the solution. Indeed the resonant modes, can be computed in a first step from a generalized eigenvalue problem and can be used in a second step to give an expansion of the solution. For unbounded domains like radiation or scattering problems the fields cannot be expressed with resonant modes of the structure. Padé techniques fall into two categories : either they generate a reduced order model directly from the matrices or they give an approximation of the frequency response from a set of full computations over the band. Recently reduced order models addressed radiation problems in the 3D case [4-6].

In this work Padé rational approximations built from a small number of full wave computations are used for solving radiation problems in three dimensions (3D). The numerical techniques are shown to provide fast broad band analysis of patch antennas.

II. wide band electromagnetic analysis

We consider the 3D problem of a radiating antenna. A computation using the FEM is performed in a finite region which includes the antenna and some of its surrounding medium. Let denote the complex matrix obtained from the discretization of the vector wave equation with lowest order tetrahedral edge elements. A first order absorbing condition is prescribed on the outer boundary. The resulting matricial system is given by :

(1)

where v is the unknowns’ vector, b is the excitation currents vector and A is a matrix which only depend on the mesh and on the medium.

The matrix can be written:

(2)

where are constant matrices

Consider such that is non-singular. By using elementary algebra the coefficients vi and bi of the Taylor series expansion of the solution vector v and right hand side b are given by :

(3)

where is the derivative of order k of .

Since from (2) appears as a polynomial of order 2 in, the number of terms appearing in the sum of (3) is limited to 2. It is very important to note that only a single inverse is needed in the iteration procedure. For each component of a Padé approximant is a rational function of two polynomials of degrees N and M respectively. The M+N+1 unknowns of are evaluated by identifying the coefficients of in the equality

(4)

III. wide band analysis of patch antennas

The ability of the proposed approach is demonstrated in the case of a rectangular microstrip patch antenna. The antenna is excited by a feed wire as shown in the figure 1. The electromagnetic analysis is performed over a broadband GHz. “Diagonal” Padé approximation (N=M), which are known to be more accurate, are used.

Fig. 1 Rectangular patch antenna

In a first step, the whole frequency band is divided in intervals such that GHz, and. In each band the center frequency is chosen in the middle. In each interval the Padé approximation is denoted . In figure 2 is shown the comparison between the standard finite element method and the Padé approach with N =3. The two curves are in an excellent agreement except at the connection between the second and third band : the efficiency of the Padé approximation can be improved by increasing N. Figure 3 shows the results corresponding to instead of . In the standard approach the finite element problem has been solved for a number of 350 frequencies. With the Padé approximation only 3 frequencies are needed. Very well approximations are obtained even if several sharp resonance peaks are included in the studied range. Five resonances appear: one near 1.57 GHz is due to the interaction between the parallel plate capacitance and the inductance of the ground wire [7]. The others are the first four operating frequencies of the microstrip patch antenna that can be obtained analytically. The difference between computed frequencies and analytical values remains smaller than 1%. A time comparison is shown in table 1 : the Padé approximation is at least 13 times faster than a standard finite element method. All the computations have been achieved on a DELL Workstation (2.8 GHz, 1 Go).

Fig. 2. Comparison between the standard FEM method

(- - - - ) and the Padé approximation (----)

Fig. 3. Comparison between the standard FEM method

(- - - - ) and the Padé approximation (----)

IV. References

[1] M. Zhang, J.F Lee, Application of the AWE Method with the 3-D TVFEM to model responses of passive microwave components, IEEE Transactions on Microwave Theory and Techniques, vol. 46, n° 11, pp 1735-1741, 1998.

[2] T. Wittig, I. Munteanu, R. Schuhmann, T. Weiland, Two-step Lanczos, Two-step Lanczos algorithm for model order reduction,, IEEE Transactions on Magnetics, vol. 38, n°2, pp 673-676, 2002.

[3] S. Bila, D.Baillargeat, M. Aubourg, S. Verdeyme; F. Seyfert, L. Baratchart, C. Boichon, F. Thevenon, J. Puech, C. Zandhi, L. Lapierre, J. Sombrin, Finite-Element Method for the design optimisation of microwave filters, IEEE Transactions on Magnetics, vol. 40, n° 2, 2004.

[4] R. D. Slone and R. Lee, Applying Padé via Lanczos to the finite element method for electromagnetic radiation problems, Radio Science, vol.35, pp 331-340, 2000

[5] J. Rubio, M.A. Gonzalez, J. Zapata, Analysis of cavity-backed microstrip antennas by a 3D finite element/segmenttaion method and a matrix Lanczos-Padé algorith (SFELP), Antennas and Wireless Propagation Letters, Vol.1, Issue 1, pp 193-195, 2002.

[6] Yu Zhu, A.C. Cangellaris, Anew finite element model for reduced order electromagnetic modeling, IEEE Microwave and wireless components letters, Vol 11, n°5, pp 211-213, 2001.

[7] J.P. Seaux, A. Reinex, B.Jecko, J.H. Hamelin, Transient analysis of space-borne microstrip patch antenna illuminated by an electromagnetic pulse, IEEE Transactions on Electromagnetic Compatibility, vol 33, n°.3 August 1991, p 224-233.