Numerical solutions of low-frequency scattering problems

1224 zyxwvuts zyxw zyxw zyx zyxwvutsrqponm IEEE TRANSACTIONS ON MAGNElTCS, VOL.28,N0.2.MARCH 1992 Numerical Solutions of Low-Frequency Scattering Problems M. Feliziani Department of Electrical Engineering University of Rome "La Sapienza" Via Eudossiana 18, 00184 Rome, Italy Abstract - Low-frequency scattering problems are analyzed through the solution of partial differential equations imposing appropriate open boundary conditions. A 2-DFEM procedure is developed to solve Helmholtz's equation in terms of the scattered electric or magnetic fields, where the forcing terms are functions of the incident fields. In the low-frequency range, the scattering is assumed to be isotropic, and the expressions of the open boundary formulations are simplified. Wave impedance boundary conditions for the scattered fields are introduced. The results obtained by utilizing different open boundary conditions in a test configuration are compared. I. INTRODUCTION The numerical computation aspects of low-frequency scattering problems have not been extensively analyzed. They are very interesting where eddy currents, shielding and electromagnetic interferences studies are concerned. A numerical study is developed here to evaluate the response of a system excited by a low-frequency plane-wave field. Applying the scattering formulation, Helmholtz's equation is expressed in terms of the scattered electric or magnetic fields, where the forcing terms are functions of the incident plane-wave fields. For two-dimensional configurations separating the incident wave-field in vertically and horizontally plane-wave fields (E-wave and H-wave), a scalar Helmholtz's equation is derived , which is solved by a finite element procedure. Scattering problems in exterior regions have to satisfy the radiation condition at infinity, whilst partial differential equation solutions require a bounded region. The region is therefore limited and the infinity condition is replaced by open boundary conditions to assure the regularity of the elliptic equations. This problem has been successfully investigated by many authors for resolving the Helmholtz or the Laplace equations [1-41. Manuscript received July 7, 1991. However, the well-known open boundary techniques work well when free-space wave propagation is negligible, and thus the field is assumed to be in static conditions, or in the high frequency range, when the FEM mesh is truncated at a distance from the scatterer of the same order of magnitude as the wavelength [3-91. In many applications, wave-propagation has to be included in the FEM formulation, but it is not possible to limit the FEM region at a distance comparable with that of the wavelength, because it can be too large. For low-frequency scattering problems, the scattered field is isotropic [lo], and this propriety is utilized to review the absorbing boundary condition (ABC) formulation and introduce free-space impedance boundary approximations. The aim of the paper is to reconsider the open-boundary techniques for application in the intermediate-frequency range. FEM implementation of the open boundary conditions is presented. The results obtained by utilizing different open boundary conditions are compared in a test configuration, zyx II. MATHEMATICALFORMULATION zyxw A. Numerical Solution of 2-0 Scattering Fields As shown in Fig. 1 an electromagnetic wave is incident to a scatterer characterized by complex electrical and geometrical configurations. The z-independent incident fields are respectively given by: Hi = Hie -jkx uz E' = -jkx uz (la) (W where Ef and Hi are the electric and magnetic time-harmonic fields, k the wave number. The total fields EZand HZsatisfy the scalar homogeneous Helmholtz equation as follows: VzO + kzO = 0 0018-9464/92$03.000 1992 IEEE (2) zyxwvuts zyx zyxw 1225 B. Low-Frequency Scattering Fiekh zyxwvu zyxw zyxw zyxwvuts The analytical solutions of the scattered field by a perfectly,conductive cylinder of radius a excited by an E-wave (Ei=e-jk") is given in cylindrical coordinates r, q5 by [lo]: bnj-"H,c2'(lcr) e jag E: = where the coefficient bn are given by: I 4 bn = ES,HS - J,(ka)/Hn@)(ka) (9) zyx For low frequency scattering problems, the series of eqn.(8) is arrested at zero term, and the dominant coefficient is bo = - j n / 2 log(ka) (10) The scattered electric field is given by [lo]: El = - (2jIkn) e-jb/rIn Cj7d4 log(ka)] Fig.1. Scatterer in a low-frequency field (11) Eqn.(l 1) demonstrate that the low-frequency scattering is undependent on the coordinate $, and therefore it is assumed to be isotropic. The scattered magnetic field is also isotropic [ 101. zyxwvutsrqpo and the radiation condition at infinity C. Absorbing Boundary Conditions Applying the scattering formulation the total field iP is given by: cp = cpi + cps (4) and from eqn.(2) non homogeneous Helmholtz equation is derived as follows: where After some mathematical manipulations the weak form of the wave equation is obtained as follows [8]: At a fixed frequency, the accuracy in the field computation depends on three factors [3-73: 1) distance at which the artificial boundary is applied: the distance should be fixed as a decreasing function of the frequency; 2) order of ABC formulation; 3) node (element) density. Conditions 1) and 2) are related: it is possible choose a distant outer boundary so that the solutions obtained by using first or second order absorbing boundary conditions differ by less than 3 5%. In this way the solution obtained by the second order ABC is accurate. For cylindrical shaped scatterers it is convenient to utilize the Bayliss-Turkel expression; the generic field CP is given by : awan = a w a r = (V@VW - kz @W) dQ (7) = I T F d Q which is integrated by a finite element procedure. + B awa92 (12) where a=a(r) and 8=D(r) are explicitly reported in [3]. For complex geometrical configurations the EngquistMajda approximation is often adopted [4]: (13) swan = -jk+ + (1/2jk) a w a 7 2 where 7 is the tangential. 1226 zyxwvutsrqp For very low frequency fields, the value of the electric source wave impedance is very high (Z, >> 377 n) and the value of the magnetic source wave impedance is very low (Z, << 377 0). In the intermediate frequencies an approximate value of Z, can be obtained as the value of the wave impedance associated to a radiating small dipole configuration. At a fixed distance r, Z, is derived by the ratio between the transversal components of the electric and magnetic field, whose analytical expressions are reported everywhere. The wave impedances for electric and magnetic source fields as a function of the source distance are reported in Fig.3. This approach gives accurate results when the scattering is isotropic. zyxwvutsrqpon zyxwvutsrqp III. APPLICATION A perfectly conductive cylinder illuminated by a plane wave field is examined. The incident field is considered to be an E-wave since the analytical solutions are known [lo]. The incident field is assumed to be El= 1 Vlm and Dirichlet boundary conditions are imposed on the conductor surface [121. An unvarying FEM mesh is adopted in the calculation runs as shown in Fig.4. The radius a of the cylinder is taken as r= 1 m m and the artificial boundary is fixed at a distance R=20 mm from the cylinder surface. The map of the scattered electric field at frequency of 100 MHz calculated by applying the simplified Bayliss-Turkel absorbing boundary conditions is shown in Fig.5. The electric field computed by utilizing approximate freespace wave impedance is shown in Fig.6. The differences between the two fields are in the range of 1-5%. The discrepancies increase in the zone close to the boundary. In future works a sensitivity analysis will be performed. zy zyxwvuts zyxwvutsrq zyxwvutsr zyxwvutsrqpo Pig.2.Attificial boundary with the shape of a cylinder. For low-frequency fields, the wave number is very small (k+O) and the scattering is considered isotropic [lo]. Eqns.(lZ) and (13) are simplified by neglecting the second order derivatives, building an artificial boundary which has the shape of a cylinder as shown in Fig.2. The contour integral of eqn.(7) is modified as follows: in which 6 depends on the applied approximations [3-4]. D. Free-SpaceImpedance Boundary Conditions The free space impedance Z, is the ratio between the electric and magnetic field components. Assuming an electric field source, the wave impedance at a distance r much greater of the considered wavelength X (r >> A) is given by: which is known as the far field impedance. The same result is achieved for a magnetic field source. The near field impedance depends strongly on the source type (electric or magnetic) and geometrical configuration. The idea is to introduce in the proposed formulation freespace wave impedance boundary conditions for the scattered field. They can be easily implemented in a FEM procedure since their formulation is equivalent to the well-known impedance boundary conditions [113. *-----_-_NEAR FIELD------I 10 0.06 FIELD---FIELD--- +--FAR I I I 0.1 0.5 1.0 DISTANCE FROM SOURCE NORMAULED TO I 1.0 A /2r Fig.3. Wave impedance as a function of the source distance. zy zyxwvu 1227 IV.CONCLUSIONS Low-frequency scattering problems have been analyzed in the frequency domain through the numerical solution of the Helmholtz equation. Open-boundary techniques for applications in the intermediate-frequency range have been examined. Since the low-frequency scattering is isotropic, a simplified expression of the absorbing boundary conditions has been derived. Approximate free-space impedance boundary conditions have been investigated. The results obtained by utilizing different open boundary techniques give useful indications for the analysis of low frequency scattering problems. REFERENCES Fig.4. Mesh adopted in the calculations. P.P.Sylvester, D.A.Lowther, C.J.Carpenter, E.A.Wyatt,"Exterior finite elements for 2-dimensional fields problems with open boundaries", Proc. of IEE, Vo1.134, No.12, 1977, pp.1267-1270. O.C.Zienkiewicz, K.Bando, P.Bettess, C.Emson, T.C.Chiam, "Mapped infinite elements for exterior wave problems", && Numer. Methods &, V01.21, 1985, pp.1229-1251. A.Bayliss, M.Gunzburger, E.'krkel, "Boundary conditions for the numerical solution of elliptic equations in exterior regions", J. 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