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Scattering of Electromagnetic Waves from Complex Conducting Objects Using a Finite-Element Near-Field Computation Method

P. Frangos a; N. Nikitakos a; N. Uzunoglu a a Department of Electrical and Computer Engineering and Electroscience Division, National Technical University of Athens, 15773 Zografou Athens, Greece Online Publication Date: 01 March 1997 To cite this Article: Frangos, P., Nikitakos, N. and Uzunoglu, N. (1997) 'Scattering of Electromagnetic Waves from Complex Conducting Objects Using a Finite-Element Near-Field Computation Method', Electromagnetics, 17:2, 185 - 197 To link to this article: DOI: 10.1080/02726349708908527 URL: http://dx.doi.org/10.1080/02726349708908527

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SCA'ITERING OF ELECTROMAGNETIC WAVES FROM COMPLEX CONDUCTING OBJECTS USING A FINITE-ELEMENT NEAR-FIELD COMPUTATION METHOD

P. Frangos. N. Nikitakos, and N . Uzunoglu National Technical University of Athens Department of Electrical and Computer Engineering Electroscience Division 9, lroon Polytechniou Str., 15773 Zografou Athens, Greece

ABSTRACT A novel method for the calculation of scattering of electromagnetic waves from closed conducting objects of arbitmy shape is presented. According to this method, the surface of the conducting scatterer is divided to a finite number of plane quadrilateral patches, on each of which the electric field boundary condition is applied. An inhomogeneous linear system of equations for the unknown current coefficients on the scatterer's surface is subsequently derived, which is solved by using traditional numerical techniques. The far - field scattering diagram is subsequently calculated numerically. Finally, numerical results for the case of a cubic scatterer are provided. 1. INTRODUCTION

The problem of scattering of electromagnetic (EM) waves from objects of arbitrary shape, both conducting and dielectric, has attracted the interest of researchers during the past decades (R.F. Harrington 1968; C.A. Balanis 1989; M.M. Ney 1985; A.W. Glisson and D.R. Wilton 1980; S.M. Rao et. al. 1982; D.H. Schaubert et. al. 1984; G.E. Antilla and N.G. Alexopoulos 1994; J.L. Volakis et. al. 1994). For this purpose, methods like the Electric Field Integral Equation (EFIE) Method and the Electromagnetics. 17:185-197. 1997 Copyright 0 1997 Taylor & Francis 0272-6343197 $1 2 . 0 0 + .OO

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Magnetic Field Integral Equation (MFIE) Method (C.A. Balanis 1989; A.W. Glisson and D.R. Wilton 1980; S.M. Rao et. al. 1982; D.H. Schaubert et. al. 1984; G.E. Antilla and N.G. Alexopoulos 1994) have been used, in conjunction with the use of the Method of Moments (MOM; see R.F. Harrington 1968, C.A. Balanis 1989, M.M. Ney 1985). Here we present an alternative novel method for the problem of scattering of EM waves from conducting objects of arbitrary shape. P.V. Frangos et. al. (1993) considered and solved the corresponding to the present work homogeneous problem, in which the complex natural frequencies of conducting objects of arbitrary shape were calculated (the general approach used there will also be used in the present work). In that formulation (P.V. Frangos et. al. 1993) the surface of the conducting object is divided to a finite number of plane quadrilateral patches, and the current at each patch is assumed constant (this corresponds to rectangular basis function, in the context of the MOM; see C.A. Balanis, 1989). Subsequently, at each patch the two components of the electric current along the two diagonals of the quadrilateral patch are considered, and the electric field boundary condition is applied at all patch centers (P.V. Frangos el. al. 1993; therefore a point - matching technique, see R.F. Hanington 1968, C.A. Balanis 1989 and M.M. Ney 1985). Next, the electric field at each patch center, considered as the 'observation point', is calculated (P.V. Frangos et. al. 1993), through the concept of the vector (magnetic) potential and the use of the superposition principle, in terms of the patch current coefficients (these electric currents are located at the 'source points' of the object's surface). The calculation of the near electric field, described above, is performed either approximately, for the case of distant patches, or in an exact analytical manner for the case of near patches (P.V. Frangos et. al. 1993), through the method introduced by K. Mahadevan and H.A. Auda, 1989 . If the matrix of the linear homogeneous system is denoted by G , then the electric field calculation mentioned above directly yields the values of the matrix elements Gij . Note that in this linear system of equations the unknown vector is the current coefficients vector. Finally, the complex natural frequencies of the conducting object are calculated as the frequencies at which the determinant of matrix G equals to zero (P.V. Frangos et. al. -

'

' This method was recently extended by the authors (P.V.Frangos et.al. 1996) for the case of trapezoidal or triangular patches of uniform distribution of current.

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1993). In that reference the above method was applied for a conducting object of cubic geometry. It was found there that it is particularly appropriate for the resonance region, i.e. for ka 1 , where k=2xA is the wavenumber and a is the length of the cube's side. Furthermore, a variety of accuracy tests (P.V. Frangos et. al. 1993), such as the calculation of matrix elements Gij with different methods, as well as the comparison of the calculated natural frequencies along the real frequency axis with well - known analytical results, validated the accuracy of the proposed method. In this work we deal with the corresponding inhomogeneous problem, in which the source vector for the linear system of equations corresponds to the incident EM wave, exciting surface currents on the scatterer's surface. Note here that the matrix G , mentioned above, is the same to that calculated for the corresponding homogeneous problem (P.V. Frangos et. al. 1993), as it will be explained below (Section 2), a fact which greatly moderates the computational effort for the scattering problem considered in this work. In Section 2 (Problem Formulation) the proposed method, as outlined above, is presented. Section 3 presents numerical results of the proposed method for a scanerer of cubic geometry. Concluding remarks and future directions of this research are presented in the final Section (Conclusions).

-

2. PROBLEM FORMULATION Consider the scattering geometry of fig. l(a), in which the incident EM wave is assumed to be polarized parallel to the z - axis, while it travels along the positive y direction :

where exp(@) time - dependence is assumed and j = f i . If we assume that the scatterer's surface is divided to N plane quadrilateral patches, then the electric current induced on the i-th patch (symbol i denotes 'source' points), assumed constant, is decomposed into two components, along the direction of the two patch diagonals (P.V. Frangos et. al. 1993):

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fii4

where fii3 and are unit vectors along the i-th patch diagonals [fig. l(b)]. The electric field boundary condition states that for all patch centers (j=l to N , where j denotes an 'observation' point ), i.e. the cross - points of the patches' diagonals, the total electric field should be normal to the conducting surface at the particular point :

(Einc + ~ ~ ) . l ; r , J ~ = o (j=l to N)

inc + ~ ~ ) . f i i ' ,= o ( where is the scattered electric field calculated on the scatterer's surface, at the cross - diagonal point of patch j . Following the method introduced by P.V. Frangos et. al. (1993), the scattered electric field E ~ ( L )at an arbitrary point outside the scatterer can be expressed, by superposition, as the sum of the electric fields :?I (L) due to the current induced onthe i-th patch (i=1,2, ...,N) :

In terms of the corresponding vector (magne!ic) potential A:(!),

the

r ) given by (P.V. Frangos et. al. scattered field components -I~ ~- ( are 1993) :

where

Here one should not confuse symbol j, referring to an observation patch on the surface of the scatterer, with symbol j (italic) denoting f i [see Eqn. (I)].

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I@)

FIGURE 1 . (a) Scattering of a linearly polarized plane EM wave from a conducting scatterer of arbitrary shape. (b) Division of the surface of the complex scatterer to a finite number of plane quadrilateral patches. The diagonals of the i-th surface patch are also shown. (c) Scattering of an EM wave from a cubic scatterer. Here the surface of the cube is divided to rectangular patches of equal area. The i-th patch represents the source patch (with position vector r'), while the j-th patch is the observation patch (with position vector 5).

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From Eqns. (3), by using Eqns. (4) - (5), we obtain :

where ):( are given by Eqn. (6). Eqns. (7), in conjunction with Eqn. (6), represent a linear inhomogeneous system of equations for the

J: and 1; , where the number of equations (2N) equals the number of the unknown current coefficients. Then, we write this system of linear equations in the following symbolic form : unknown current coefficients

where J is the current coefficients vector (unknown), E'"'.~' is the vector associated with the projections of the incident electric field (local values) along the unit vectors

klJ3 and

~i~ of the j-th

patch, and

G is the connecting matrix. Note that the corresponding to Eqn. (8) homogeneous system of equations has already been considered and solved (for the case of cubic geometry) by P.V. Frangos et. al., 1993. In that reference, the matrix elements Gij , related with the calculation of the near EM field at an observation patch j due to a uniform current distribution at a source patch i, as seen from Eqns. (5) - (7), were carefully calculated, using either approximate methods (for the case of distant patches) or an exact analytical method introduced by K. Mahadevan and H.A. Auda, 1989 (for the case of near patches). Furthermore, a comparison of the computed values Gij through the two different methods mentioned above showed the accuracy of calculations (P.V. Frangos et. al. 1993). Once the current coefficients J:

and

J; are calculated, from the solution of linear system (S), where standard numerical techniques are used for the system solution (S.D. Conte and

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C. De Boor 1980; W.H. Press et. al. 1986), the scattered far field is easily calculated as follows :

The implementation of the scattering problem for the case of a scatterer of cubic geometry is now in order.

3. IMPLEMENTATION FOR THE CASE OF A SCATTERER OF CUBIC GEOMETRY - NUMERICAL RESULTS For the case of a scatterer of cubic geometry [fig. I(c)], there exist only six (6) distinct directions for the unit vectors fi13 and along the patches' diagonals. For the faces parallel to the xOy plane, these vectors are given by :

fii4

(Similar formulae are found for the faces parallel to the xOz and yOz plane, see N. V. Nikitakos 1996). The linear system corresponding to Eqns. (6) and (7) [or Eqn. (8)] is now formulated. The left-hand-side (LHS) of Eqns. (8) has already been formulated by P.V. Frangos et. al. (1993); the reader should consult with G . that reference for details regarding the formulation of matrix Regarding the right-hand-side (RHS) of Eqns. (7), we will consider both the cases of wave incidence normal to a cube's face [the face parallel to xOz plane, in particular, see Eqn. (I)], and obliquely incident upon a cube's face. For the latter case, if we assume that both the wavevector k and the electric field vector of the incident wave have, without loss of

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generality, only y and z components, the incident electric field can be written in the form [fig. I(c)] : Einc = Eo (cose. i + sine. i) exH-j

-

h)= ~,

where 8 is the angle of incidence and j = f i [for 8=Oo, Eqn. (1) is recovered as a special case]. Calculating the inner products of the R H S of Eqns. (7) according to Eqns. (10) - (1 1) [and the analogous equations for the xOy and yOz planes], we obtain, for the case of faces parallel to the xOy plane: Xy

Einc

" " -

XY

-N24

.-Einc

1

C

,

face z = 0

*%sin0ex~-~k~~,cos01 x p ( j k .asin@ , face z = a

(12) Note in Eqn. (12) that the k cases correspond to the two cases of the LHS of this equation, while the appropriate phase factor at the RHS of the equation is chosen from the face of the cube that we refer to, i.e. face z=0 or face z=a [Expressions similar to that of Eqn. (12) are derived for the inner products at the R H S of Eqns. (7) corresponding to the faces parallel to the xOz and yOz planes ; see N.V. Nikitakos, 1996, for more details]. Numerical results are displayed in figs. 2 and 3. Fig. 2 corresponds to the case of normal incidence, Eqn. (1). Figure 2(a) shows the far - field scattering pattern in the xOy plane, and fig. 2(b) the scattered far field in the yOz plane. The plots shown are for values ka = 3.14 ( a l l = 0.5 , solid lines) , and ka = 6.28 ( a l l = I , small circles line symbol). Calculations have been performed by dividing the surface of the cube to N = 96 rectangular patches of equal area (order of matrix G equals to 192 x 192, see P.V. Frangos et. al., 1993) . Furthermore, a standard Gauss - Jordan elimination technique (W.H. Press et. al. 1986) was used for the numerical solution of the linear system (8), while the calculation of the scattered electric field was performed at a distance R =400h

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FIGURE 2 . Far - field scattering pattern in the xOy plane (a) and yOz plane (b) for the case of normal incidence of an EM wave upon a cubic scatterer. Here a is the length of the cube's side, h, is the wavelength of the incident EM wave, and k, = 2n/h, is its wavenumber.

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FIGURE 3 . Far - field scattering pattern in the xOy plane (a) and yOz plane (b) from a cubic conducting scatterer at oblique incidence, 0 = 30" [see Eqn. (1 1) and figs. I(c) and 3(b)]. Note that in this case the symmetry of the scattering diagram is preserved only in the xOy plane (see text for details).

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(R>>X) . Because of obvious symmetry considerations, the plots are symmetric with respect to the 8=0° axis, as shown in figs. 2(a) and 2(b). Finally, fig. 3 shows the scattered far - field distribution for the case of oblique incidence [angle of incidence 8=30°, Eqn. (1 l), as shown in figs. l(c) and 3(b)]. It is obvious that, because of fundamental symmetry considerations, the scattering pattern is here symmetric only in the xOy plane [fig. 3(a)], while symmetry is destroyed in the yOz plane, as shown in fig. 3(b) [field values in the scattering plots of figs. 23 have been normalized].

4.CONCLUSIONS In this work we developed and presented a novel method for the scattering of EM waves from conducting scatterers of arbitrary shape. The method is characterized by conceptual simplicity, and is based on the division of the scatterer's surface to a finite number of plane quadrilateral patches and the application of the electric field boundary condition at the centers of these patches (point - matching technique). This leads to an inhomogeneous system of linear equations for the unknown current coefficients on the scatterer's surface. The corresponding homogeneous system of equations has been considered and solved by the authors elsewhere (P.V. Frangos et. al. 1993), for the calculation of the natural frequencies of complex objects. In that case, accuracy tests as well as comparisons with well - known analytical results3 (P. V. Frangos et. al. 1993) had validated the accuracy of the proposed method. In the present work we deal with the corresponding inhomogeneous system of equations, the source vector being related with the electric field incident upon the scatterer and exciting electric currents on its surface. Numerical results for the case of a cubic scatterer are presented. The proposed method can be run even on a mini computing system, with reasonably small execution time4 . Future directions of this research include minimization of the computer execution time associated with the

amel el^, the well - known eigenvalues of a rectangular cavity were accurately predicted by the proposed method (P.V. Frangos et. al. 1993). 4 In our case, the implementation of the proposed method for the cubic scatterer geometry that we used was performed on a HP - 90001730 computing facility (workstation), with an average of I (one) hour execution time for each of the plots shown at fig. 2 or fig. 3. Furthermore, note that the computer executions of the corresponding homogeneous problem (P.V. Frangos et. al. 1993) have been performed on a HP - 420 mini system.

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proposed method by taking advantage of the symmetries in the scatterer geometry or by using parallel computation techniques, comparison of the method performance with other techniques, previously given in the literature, and consideration of more complicated scattering geometries.

ACKNOWLEDGMENT One of us (PF) would like to thank Prof. D.L. Jaggard, Moore School of Electrical Engineering, University of Pennsylvania, USA, for useful discussions regarding this research. Furthermore, the authors would like to thank the Editor, Prof. N.G. Alexopoulos, for constructive remarks. This work has been supported in part by the NATO Collaborative Research Grant No. 930923.

REFERENCES [I] R. F. Harrington, Field Computation by Moment Methods, New York: The Macmillan Company, 1968. [2] R. F. Harrington, 'Origin and development of the Method of Moments for field computation', IEEE Ant. Propag. Magazine, Vol. 32, No. 3, pp. 3 1-36, 1990. [3] C. A. Balanis, Advanced Engineering Electromagnetics, New York: J . Wiley and Sons Edit. Company, 1989. [4] M. M. Ney, 'Method of Moments as applied to Electromagnetic Problems', IEEE Trans. MTT, Vol. MTT-33, No. 10, pp. 972 - 980, 1985.

[5] A. W. Glisson and D. R. Wilton, 'Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces', IEEE Trans. Ant. Propag., Vol. AP-28, No. 5, pp. 593 - 603, 1980. [6] S. M. Rao, D. R. Wilton and A. W. Glisson, 'Electromagnetic scattering by surfaces of arbitrary shape', IEEE Trans. Ant. Propag., Vol. AP - 30, NO. 3, pp. 409 - 418, 1982.

[7] D. H. Schaubert, D. R. Wilton and A. W. Glisson, 'A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped

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inhomogeneous dielectric bodies', IEEE Trans. Ant. Propag., Vol. AP 32, NO. I, pp. 77 - 85, 1984. [8] S. M. Rao et. al. , 'Electromagnetic radiation and scattering from finite conducting and dielectric structures : surface / surface formulation', IEEE Trans. Ant. Propag., Vol. 39, No. 7, pp. 1034 - 1037, 1991. [9] G.E. Antilla and N.G. Alexopoulos, 'Scattering from complex three dimensional geometries by a curvilinear hybrid finite - element - integral equation approach', J Opt. Soc. Am. A, Vol. 1I, No. 4, pp. 1445 - 1457, 1994. [lo] J.L. Volakis, A. Chatterjee and L.C. Kempel, 'Review of the finite element method for three - dimensional electromagnetic scattering', J Opt. Soc. Am. A, Vol. 11, No. 4, pp. 1422 - 1433, 1994.

[I I] P.V. Frangos, N.V. Nikitakos and N.K. Uzunoglu, 'Calculation of the natural frequencies of complex objects using a finite - element near field computation method', J of Electromagnetic Waves and Applications, Vol. 7, No. 12, pp. 1633 - 1652, 1993. [I21 K. Mahadevan and H.A. Auda, 'Electromagnetic field of a rectangular patch of uniform and linear distributions of current', IEEE Trans. Ant. Propag., Vol. 37, No. 12, pp. 1503 - 1509, 1989. [13] P.V. Frangos, N.V. Nikitakos and N.K. Uzunoglu, 'Electromagnetic field of a plane trapezoid of uniform distribution of current', Electromagnerics, N. Alexopoulos, Editor - in - Chief, Vol. 16, pp. 145 163, 1996. [I41 N.V. Nikitakos, 'Elecbomagnetic Methods for Radar Target Identification', Ph. D. dissertation, Department of Electrical and Computer Engineering, National Technical University of Athens, Greece, 1996, Chapter 4. [IS] S. D. Conte and C. de Boor, Elementary Numerical Analysis, 3rd Edition, New York: McGraw - Hill, 1980, Chapter 4. [16] W. H. Press et. al., Numerical Recipes, Cambridge, United Kingdom: Cambridge University Press, 1986.