A general infinite element for terminating finite element meshes



A General Infinite Element for Terminating Finite Element Meshes in Electromagnetic Scattering Prediction
A. Charles, M. S. Towers and A. McCowen
Department of Electrical and Electronic Engineering, University of Wales Swansea. UK
Abstract - In this paper a general infinite element is presented which can be used in the termination of finite element meshes for scattering electromagnetic problems. The new infinite element uses an increased number of nodes in the radial direction which provides it with much improved absorption over the previously presented infinite elements. This paper will demonstrate that the previously reported two noded and four noded infinite elements are inferior in performance compared to the Bayliss Turkel absorbing boundary condition and that the multi-noded infinite element is able to give comparable results to the Bayliss Turkel ABC. Index terms - Electromagnetic scattering, Finite element methods, Radar cross sections, Modelling, Infinite elements.

to allow the evanescent fields to decay before they can accurately absorb the out-going wave. Furthermore, as the size of the target is increased the width of the finite element mesh needed to accurately model the fields increases. Results will also be given which show that the use of more than four nodes in the infinite element improves its performance again so that it can be compared with the Bayliss Turkel ABC in terms of accuracy, for a given size of finite element mesh.

The technique of finite element analysis lends itself well to the modelling of electromagnetic waves and is of particular interest in investigating the scattering from general bodies with regions of inhomogeneous material. In general the solution to the scattering problem is unbounded and therefore a method of terminating the finite element mesh is required to absorb the scattered waves. In recent years there has been much attention given to absorbing boundary conditions (ABC’s) [I]. A popular and well documented ABC is the Bayliss Turkel ABC [2], which employs the Wilcox expansion, neutralising the first few terms of the series. The Bayliss Turkel ABC is often used as a standard by which many other ABC’s are measured, and will be used as a benchmark in this paper. There has also been some interest in the use of the infinite element [3,4] as an alternative means of truncating the finite element mesh. In recent papers four noded [5,6] and two noded infinite elements [7] have been presented. The two noded infinite element presented by Gratkowski has two nodes on the finite/infinite element boundary, coincident with those of a finite element and two nodes at infinity, but the latter are inactive and can therefore be ignored. The four noded element again has two nodes on the finitehnfinite element boundary and two nodes at infinity which can again be ignored, but also has two extra nodes a set distance out fiom the finite element mesh. This paper will give a comparison of analytic and numeric results for 2-D cylinders and show that four noded infinite elements terminate the finite element mesh better than the two noded elements. The four noded infinite elements still require a large finite element mesh
Manuscript received November 3, 1997

Shown in Fig. 1 is the construction of a multi-noded infinite element, where all infinite elements are generated from a common centre C, and have two nodes coincident with those of the abutting finite element. The parameter kl determines the location of the decay centre D a distance (1-kl)R from C along the line to P and k2 determines the position of the n* node in the infinite element, kl and k2 are both optimised in the range 0 to 1 for a particular problem. To Infmity 4nfinite element


/ 6 I



Fig. 1 . Construction of infinite elements with 2 x n nodes For all the infinite elements discussed in this paper, fields H are linearly interpolated in the circumferential direction, but interpolation in the radial direction is dependent on the number of active nodes, e.g. for a four noded infinite element radial interpolation is quadratic, and is described by the shape functions N, expressed in terms of local variables, ([,q), such that


0 1998 IEEE


converge. Equation (5) is expanded using ( 6 ) and any (1 - s)' terms cancelled so that there are no unwanted

Ej is the nodal solution variable controlling the value of
field at the infinite element node i, and ~ ~ ( 6 , " ) ithe s corresponding Lagrange shape function. k is the fi-eespace wave number. r is the position vector fiom C to the point P(E,,q) and similarly r' = r - (1- k1)R is the distance fi-om the decay centre D to P. The elements are constructed such that the nodes are evenly spaced with respect to 5 in the local co-ordinate system. i.e. (3) where n is the number of nodes along one edge of the infinite element. Therefore in the global co-ordinate system, distances fiom C are mapped according to

poles at infinity (where 5=1). Hence



z :yT 1










The element matrices are defined in the following way
Ke, =


J J(VN"VN-k*N*N)&@%I



N is a row vector of global shape functions as given in (2) and * denotes the complex conjugate transpose.



to the

Fig. 2. Local co-ordinate system of infinite elements The Jacobian J used to transform the operator local co-ordinate system is defined as

where b = 1- 4 + k , 4 and vectors R and W are defined in Fig. 2. Thus


M is the row vector of element Lagrange shape functions M&q) and MT is'the transpose. Careful examination of the Kel matrix is required to ensure that the integrals

The ability to place an ABC on a boundary conformal to the scatterer is paramount in reducing the size of the finite element mesh for long thin bodies, the formulation


b ( W .W ) 4k,lR x W I

k 2 k 1 ( RW)' . b(R x W I


for infinite elements allows this action, as does the Bayliss Turkel ABC. The optimisation of the parameters associated with the multi-noded infinite elements is discussed in this section. There are four parameters to be considered for optimising the elements' performance, not including their distance from the scattering body. The first is the number of nodes, 2x1, where it has been found that ten noded infinite elements give a satisfactory performance for all size bodies although fewer nodes may be considered when modelling smaller bodies, in consideration of computational efficiency. The second parameter is kl whch has been found to be optimum at k, = 1.0 for all problems and boundary shapes tested so far. The third parameter k2 was found to have no effect on results, since all the corresponding shape function sets span the same polynomial space used to approximate the field distributions. The final parameter is the position of the scattering centre, C, withn the body, which has been found to have very little effect on the accuracy of results.

a PEC ogival cylinder. where the two noded infinite element results is taken directly from [7]. The ogival cylinder has major and minor axes of IA,, and 0.2b. respectively and is illuminated with a TM mode incident wave at an angle of 22.62' to the major axis. The result from [7] employs two noded infinite elements placed on a rectangular boundary at least 2 . 8 b from the scatterer, whereas the ten noded infinite elements are placed on a conformal boundary 0.33ho from the body. The results from the ten noded infinite element can be assumed to be correct because the farfields from surface currents and from infinite elements are indistinguishable in Fig. 3. In addition to the result being more accurate. the finite element mesh in the ten noded infinite element problem is much smaller and therefore results in a much faster solution time.

There are two methods of calculating the Radar Cross Section (RCS) of a scattering body, which will be discussed here, namely farfields from surface currents and farfields from infinite elements, both presented in [ 5 ] . The derivation of farfields from surface currents is indistinguishable to that given in [ 5 ] , but a small change needs to be made to the expressions for farfields from infinite elements. The new expression includes the parameter klso that


.... 000

result h m [7] 2 noded infinite element farfield fbm surface currents 10 noded infmte element farfield &ominfiate elements 10 noded infmte element

Fig. 3. Bistatic RCS from PEC ogival cylinder in TM mode where

lH,ncl is the magmtude of the incident wave.
I $

L =l h ( ~ , ) , and

is measured anti-clockwise from the

positive x-axis.

Thus ~ ~ ~ can4 be easily found from the nodal ( ) values on the infinite element. When this RCS value is
compared with the RCS calculated from surface currents it is thought that the degree of agreement gives an indication of the accuracy of the result. In all cylinder problems tested so far if the two values agree well the RCS has been found to be close to the correct value. In these cases the finite element mesh was checked independently for accuracy by applying the analytic nearfield to its outer boundary as a termination. Thus the errors seen using the infinite elements are due entirely to the quality of termination Shown below is a series of results demonstrating the improvements which can be made if the number of nodes on the infinite elements is increased. Fig. 3 provides a comparison of two noded and ten noded infinite elements. used to model the bistatic RCS of



Fig. 4.RCS from 4hodiameter PEC cylinder in TE mode The results in Fig. 4 give a comparison between four, eight and ten noded infinite elements terminating identical finite element meshes 0 . 3 3 b from a PEC circular cylinder. The problem is solved for the TE mode


and the result for the ten noded infinite element is indistinguishable from the analytic solution. Both these problems show how the size of the finite element mesh can be reduced if more nodes are used in the infinite elements.

monostatic RCS for the same problem, this result however shows the infinite elements to be better than the Bayliss Turkel ABC, thus in general it can be said that the multi noded infinite elements are comparable to the Bayliss Turkel ABC. For both problems an elliptical boundary was fitted to surround the wedge no closer than 0.5ho. For the same mesh and boundary the four noded infinite elements give significant errors compared to the result in Fig. 5, of up to 30 dB.



.. -.

..- Eayliss Turkel ABC
lnfinlte Elements -50

0 (I)








Angle (degrees)

Bayliss Turkel ABC

Fig. 5. Bistatic RCS fi-om PEC wedge in TE mode






Angle (degrees)

Fig. 7. Monostatic RCS fi-om PEC wedge in TE mode







A new general multi-noded infinite element has been presented in this paper. Comparison has been made between the previously presented infinite elements, the Bayliss Turkel ABC and the new multi-noded infinite elements. The results clearly show that the new multinoded infinite elements are able to give improved results over the infinite elements presented so far and are now similar in performance to the Bayliss Turkel ABC.

Angle (degrees)

Fig. 6.Error in PEC wedge bistatic RCS problem
An example is also given to show the performance of the multi-noded infinite elements in comparison to the Bayliss Turkel ABC Fig. 5 nhown the bistatic RCS &om a PEC wedge, in TE mode, the dimensions of which are shown inset. The 'accurate' result here is obtained by solving the problem with a fine mesh and a large white space (i.e. the area between the body and the ABC), where the infinite elements are placed 1.75ho from the scattering body. These results show that the ten noded infinite elements perform as well as the Bayliss Turkel ABC even when they are brought close to the surface of the scatterer. Fig. 6 shows the error of the Bayliss Turkel ABC and the infinite elements. The distributions appear remarkably similar and are thought to be due more to the finite element mesh than to the termination. Fig. 7 shows the

R Mittra, 0 Ramahi, A Khebir, R Gordon, A Kouki, "A review of absorbing boundary conditions for two and three dimensional
electromagnetic scattering problems" IEEE Trans on Mag, vol 25, no 4, pp 3034-3039,1989 A Bayliss, E Turkel, "Radiation boundary conditions for wavelike equations" Comm on Pure and Applied Math vol 33 pp 707-725, 1980 P Bettess, C Emson, T C Chiam, "A new mapped infinite element for exterior wave problems" Numerical methods in coupled systems, 1984 P Bettess, "Infinite elements" Intemational joumal for numerical methods in engineenng, vol 11 , pp 53-64, 1977 J Macnab, A McCowen, M S Towers, "Electromagnetic scattermg fkom an arbitrary inhomogeneous 2-D object - a finite and infinite element solution" IEEE Trans on Ant and Prop, vol 41, no 6, pp 770-777, 1993 J MeDougall, J P Webb, "Infinite elements for the analysis of open dielectric waveguides" IEEE Trans on Microwave Theory and Tech, vol 37, no 11, pp 1724,1989 S Gratkowski, L Pichon, A Razek, "New infinite elements for a finite element analysis of 2-D scattering problems" IEEE Trans on Mag, vol 32, no 3, pp 882485,1996



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