IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 12, DECEMBER 2005
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An FEM Approach With FFT Accelerated Iterative Robin Boundary Condition for Electromagnetic Scattering of a Target With Strong or Weak Coupled Underlying Randomly Rough Surface Peng Liu and Ya-Qiu Jin, Fellow, IEEE
Abstract—To study electromagnetic scattering of a target above randomly rough surface, the Robin boundary condition (Robin BC) is employed to separate and enclose two isolate regions of the target and underlying surface. The fields in these two enclosed regions are strongly or weakly coupled by iteratively solving the scattering field integral equation in the finite element method (FEM) with updating the right-hand side residual of the Robin BC. This FEM approach presents an effective calculation for the model of the target at high altitude above large-scale rough surface to show their coupling interactions. Most time consuming of the algorithm is spent on evaluating the Robin BC on the fictitious planar boundary over the large-scale rough surface. The scattering field integral equation is written as the one-dimensional convolution form and is solved efficiently by using the fast Fourier transform (FFT). Our approach is first validated by available FEM-DDM (domain decomposition method) results. Then, the functional dependence of bistatic and back-scattering from the target above rough oceanic surface upon the target altitude, incident and scattering angle, etc., are numerically simulated and discussed. This study presents a numerical description for the scattering mechanism associated with strong or weak coupled interactions of a volumetric target at various altitudes above randomly rough surface. Index Terms—Electromagnetic scattering, finite element method (FEM), iterative method, Robin boundary condition (Robin BC), target and rough surface.
I. INTRODUCTION
E
LECTROMAGNETIC scattering of the target above randomly rough oceanic or land surfaces is one of the most important subjects for radar surveillance, target tracking and microwave remote sensing, etc. There have been many studies, e.g. experimental observation [1]–[3] and numerical simulation such as the approaches of the half-space Green’s function and the planar boundary approximation [4]–[6]. With the advancement of computational technology, numerical modeling and
Manuscript received March 31, 2005; revised June 15, 2005. This work was supported by the China State Major Basic Research Project 2001CB309400 and NSFC Project 60571050. The authors are with the Key Laboratory of Wave Scattering and Remote Sensing Information, Ministry of Education, Fudan University, Shanghai 200433, China (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/TAP.2005.859904
simulation of electromagnetic scattering from the object above or below the rough surface has become the focus of extensive attention [7]–[10]. Some primary algorithms such as the method of moments (MoM), fast multipole method (FMM), fast Fourier transform (FFT), etc., have been extensively studied to solve the electric-magnetic field integral equations (EFIE-MFIE). Bistatic scattering from a target at low altitude above rough sea surface has been discussed in [11] by using the finite element method (FEM) with the conformal perfectly matched layer (CPML). In succeeding study [12], the noniterative FEM-DDM is developed for the numerical model of a target on or above the rough oceanic surface. A two level quasistationary algorithm [13] to the FEM-DDM is also studied to obtain the Doppler spectrum of a fast flying target above temporally dynamic sea surface. As the target is close to the underlying rough surface, scattering interaction between the target and surface is strong. When the target arises, coupled interactions of the target and rough surface, as well as their specular links, generally become weak. However, the numerical FEM region to enclose both the target and surface is significantly increased when the target arises. It makes serious difficulties to carry out numerical calculations over the whole large region enclosed. The iterative Robin boundary condition (IRBC) [14]–[16] is expressed as the localized operator embodying the Sommerfeld radiation condition on the fictitious boundary, and separately encloses the multiscatterers. The difference of the IRBC from conventional Robin BC is that the right-hand side residual is not fixed during computation and is iteratively updated by solving the field integral equation. The IRBC is set to enclose the scatterers, and iterations can quickly converge to the accurate result. In this paper, the IRBC is employed to enclose the target and the underlying surface, and the FEM is developed to numerically simulate bistatic and back-scattering from the target at high altitude above randomly rough surface to show how strong or weak coupled interactions they might have. In fact, iteration times of the IRBC is reduced when the target arises and leaves from the underlying surface. Most time consuming of the algorithm is spent on evaluating the IRBC residual over the planar fictitious boundary of the large-scale rough surface. The scattering field
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small area of free space. Over the large-scale rough surface , the boundary is planar with quarter-elliptical shaped sections patched at both ends, while the boundary encloses the target with a smooth and convex structure. The electric field integral equation (EFIE) can be used as the open-region BC for the multitargets scattering problem [26], [27] as given as follows: Fig. 1. Physical model of the problem.
integral equation is written in the 1D convolution form, and the analytical derivation and numerical FFT are developed to accelerate computations. Our approach is first validated by available FEM-DDM code and results [12]. Functional dependence of bistatic and back-scattering from the target at various altitudes over an oceanic rough surface upon the target altitude, incident and scattering angle, and some other parameters are numerically simulated and discussed. II. THE FEM MODEL WITH IRBC Numerical FEM approach employs an open-region boundary condition to enclose the computation region. There are usually two kinds of open-region boundary condition, i.e., the global boundary condition and the local boundary condition [17]. The global boundary condition is numerically exact open-region BC for the FEM, such as EFIE [18] and the unimoment method [19], which relates the field at one point of the boundary to the fields at the remaining part. On the contrary, the local boundary conditions, such as the absorbing boundary condition (ABC) [20], [21] and perfectly matched layer (PML) [22], are less accurate, and can describe the radiation characteristics of the scattered fields in a localized form or the physical properties of the absorbing medium. The global and local BC’s have respective advantages and bottlenecks. For example, the global BC is accurate and can be placed close the scatterer, but produce a full matrix equation and destroy the sparsity of the final FEM system matrix. The local BC is approximate and has to be imposed at a certain distance from the scatterer, but can preserve the sparse system matrix for efficiently storing and solving FEM [17], [23]. The physical model of this paper is shown in Fig. 1: supabove a 2-D rough surface pose to have a target located at , where is respect to the mean value 0. An oceanic rough surface is generated by using the Monte Carlo method with the Pierson–Moskowitz (P–M) sea spectrum [24]. Both the target and the rough sea surface are perfectly conductors. The surface length and the target are illuminated by a tapered electromagnetic wave [25] incidence. In the FEM approach with the local BC such as the CPML [11], the local BC should enclose whole region of the target and underlying rough surface, where the enclosed free space is quickly increased when the target arises leaving from the underlying surface. Numerical computation over so large free space region remarkably reduces the effectiveness. To avoid this situation, two fictitious boundaries and are introduced to separate the computation region into two isolated and to enclose the target and underlying surface, regions respectively. As shown in Fig. 1, each region contains only a
(1) where is the scattered electric field satisfying the Sommerfeld radiation condition [17], [27] (2) is the outward normal derivative on the boundand . The 2-D free space Green’s function is 4, where is the distance between the observation point and the , and is the wave number in free space. source point is When overlaps , the integration on the boundary or evaluated in the Cauchy principal form [28]. As the target arises, the length of the rough sea surface is correspondingly increased to cover larger interaction space, and the dimension of the dense matrix generated by the IE is quickly increased. It is known that for the elongated scatter structure, enclosed by and the rough sea surface, such as the domain the FEM becomes less efficient when the IE is used as the openregion BC [12], [29]. To circumvent this difficulty, a FFT accelerated IRBC is presented as the truncation boundary for the FEM. Suppose the Robin BC on the fictitious boundaries and has the form of
and aries
(3) where is the operator of Robin BC, which is similar to the left-hand side of (2), i.e., the Sommerfeld radiation condition when the enclosed boundary apreaches the order . However, as is finite at the enclosed proaches boundaries or , the Sommerfeld radiation condition as given in (3). reaches the residual is not yet determined, an initial Because the residual guess is needed to carry out iterative Robin BC. The fields in the and , are numerically solved by the FEM with domains, the prescribed Robin BC. Then, using the fields of the integral and , the residual term on equation on the paths and is updated for next iteration. Such procedure is carried out iteratively until the fields in and converge to the accurate result. Most time consuming of the IRBC is spent on evaluon the planar fictitious boundary over the large-scale ating and are parallel straight line with rough surface. Because the distance over the surface length , the integral equation is simplified to the 1-D convolution form, which can be calculated efficiently by using FFT.
LIU AND JIN: AN FEM APPROACH WITH FFT ACCELERATED ITERATIVE ROBIN BOUNDARY
Normally, an initial guess of the residual on and is zero. Evaluating the stationary point of the following functional problems [15], [17] by the FEM
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4 1
1 2
(10) (4)
2 1 2
The residual on can be calculated similarly. The next iteration of the IRBC starts from the step of solving (4–5) with , and then the above procedure is repeated until the updated and converge to the accurate value. the fields in III. THE IRBC WITH FFT ACCELERATION
(5)
2
the scattered fields in the domains and are determined. Because of the locality of the Robin BC, the sparsity of the system matrix is maintained and the FEM can be performed with high efficiency. is Then, by applying the operator to (1), the residual updated for the next iteration, where the integral path is set on and , such as at point on
Most time consuming of the IRBC is spent on evaluating the on the planar fictitious boundary over the largeresidual scale rough surface, i.e., calculation of in (6) over . Since and are the parallel straight lines with the distance in can be written in the 1D convolution Fig. 1, the integral on form. Supposing that the planar sections of and are divided with the equal divisions, the FFT can be used to accelerate the computation.
(11)
(6) where the subscripts and represent the contribution from (s), rethe target (t) and rough surface (s) to the point on is the directional derivative in the outward spectively, and normal direction of the paths and . Because , and , are parallel as shown in Fig. 1, there is no singularity in evaluof (6). The first and second order partial differentials ating of the 2-D Green’s function are written as
Due to the large-scale rough surface of the model, the scattered fields are almost zero at the ends of the planar boundary. Therefore the contributions from the quarter-elliptical shaped sections in Fig. 1 are negligible in (11), where denotes the 1D convolution operator, , , and both and are constants. The discrete Fourier trans, , , and are , , forms (DFT) of , and , respectively, with and given by 1 4 (12)
(7)
4
and
(8)
1 4 (13)
, with
respectively, where
isreduced ByusingtheFFT,thecomputationcomplexityof in (6) to in (11), where is the number from and . of sampling points on the planar section of If the rough surface is long enough 1000 , the DFT of the Hankel function in (12) and (13) may be approximated by the analytical Fourier transform [30] in the infinite domain
4 1
(9)
2
(14)
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Fig. 2. Beam formation of the tapered incident wave ( g 68.267).
=
= 60
,
= 1 m, Fig. 3. Bistatic scattering from a target above the rough sea surface (h U 3 m/s).
where
=
(15) Due to the properties of the Fourier transform [31], the analytical Fourier transform of and are 2
(16)
(17) 2 Notice that the denominator of (17) equals zero when . This singularity may be avoided by a slight change of the length of the rough sea surface [30]. IV. NUMERICAL RESULTS AND DISCUSSIONS Applications of the IRBC in FEM for solving the scattering problem of single and multiple scatterers have been studied in [14], [16], where the IRBC is placed closely to the scatterer and converges quickly to the accurate results. The following example shows its effectiveness on scattering problem of combined target-rough surface. To eliminate the edge effect when ideal plane wave is used to illuminating the finite surface length, the Thorsos tapered wave is applied with amplitude given by [25] (18) versus when the incident angle Fig. 2 shows 60 , the wavelength 1 m, and the tapering parameter 68.267 . It can be seen that the tapered wave resembles the , while its intenplane wave near the central axis 2 . sity becomes very small when The rough oceanic surface is driven by the surface wind speed 3 m/s with the surface length 1000 . The axial ratio is 6, while the major axis of the quarter-elliptical sections of of the ellipse coincides with the horizontal plane. Other param1.0 and 0.2 . eters are chosen as We now suppose to have a target of NACA0012 four-digital airfoil [33] with the chord length 2 located above the rough 7.794 and the center surface. With the leading edge at axis at the height 4.5 , the airfoil is located at the center
= 4.5,
enof the incident tapered wave. The elliptical integral path closes the airfoil target with the major and minor axis lengths is with 2.4 and 0.8 , respectively. The fictitious boundary the equal distance 0.2 away from . Fig. 3 compares the bistatic scattering coefficient , i.e., bistatic RCS (radar cross section) obtained by our Robin-FFT and FEM-DDM [12]. The Robin-FFT approach converges within 14 iterations when 0.01 dB, where the end-iteration tolerance is set as is the maximum angular 1 . difference of bistatic RCS in adjacent iteration steps The Robin-FFT approach agrees well with FEM-DDM in all 90 , scattering directions, except near the extreme where a reflection error in the FEM-DDM is produced due to the abrupt change of the boundary (i.e., from the CPML to the conducting rough sea surface). Bistatic RCS of the target using the image theory is included, and well agrees with the numerical results except near the normal and specular directions. In order to make an observation of the scattering interference between the target and the underlying rough sea surface, the surface wind speed in the following simulations remains 3 m/s with 0.3016 when 1 m and is the rms height of rough surface. The Robin-FFT also agrees well with the FEM-DDM under strong wind speed, for example the mean 7 m/s. error is 0.66 dB when A. Bistatic Scattering as a Function of Based on the two-source array radiation and image principle, single scattering from the target and its mirror image under a as [11] plane interface have the phase difference 2
(19)
for 4.5 is shown on the upper half of where , alternatively, correFig. 3. It can be seen that sponds to the maximum and minimum of bistatic RCS, i.e., 1,3,5 2 1 for the maximum and 2,4,6 2 for the minimum, where indicates the peaks number in the 9 shown in forward and backward scattering patterns, e.g. Fig. 3.
LIU AND JIN: AN FEM APPROACH WITH FFT ACCELERATED ITERATIVE ROBIN BOUNDARY
Fig. 4. Bistatic scattering from a target above the rough sea surface (h U 3 m/s).
=
= 9.5,
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Fig. 6. Bistatic scattering of the target at different altitudes above the rough surface in strong-coupling region .
A
Fig. 5. Interference of incident and reflected waves when the target is at different altitudes.
Because 2
(20)
variation ratio of becomes increased with larger and , i.e., the angular pattern of bistatic scattering, becomes more increases when is fixed. Note that this prinfluctuating as ciple is not related with the incident angle. (c.f. the result at low 80 in [11]). grazing angle (LGA) B. Bistatic Scattering as a Function of Target Altitude 1) Angular Fluctuation of Bistatic RCS: If the target arises vertically, it might gradually move out from the tapered beam in the case of oblique incidence. In this paper, the target altitude arises along the central axis of the incident tapered wave, i.e., . the horizontal position of the airfoil is kept as Fig. 4 shows bistatic scattering when the target is above rough 9.5 . It took 12 iterations for the Robin-FFT sea surface at 0.01 dB. Comparing with Fig. 3, the flucto converge to tuation of angular scattering pattern becomes more frequent as the target altitude increases. Following (19) and (20), both and increase, and 19 in Fig. 4. 2) Fluctuation of Bistatic RCS in Strong-Coupling Region: Since the incident beam width is always finite in practical measurement, there exists a strong-coupling region as shown in above the rough sea surface, such as region
Fig. 7. Bistatic scattering of the target at different altitudes above the rough surface in weak-coupling region .
B
Fig. 5, where the incident wave and the reflected wave of the underlying surface arrive at the target with the interference . phase difference
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Fig. 8.
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Bistatic scattering of the target above the rough surface (h
= 50, L = 2000).
Under the ray optics approximation, the phase difference beand beam is given by tween the beam 2
(21)
Similar to (19), varies with increasing , and causes periodical fluctuation of the induced currents on the target surface. 4.5 As shown in Fig. 6, the bistatic RCS corresponding to is almost entirely larger than the target scattering in free space (i.e., in the absence of the rough sea surface), and the case of 5.0 is alternative. Because the interference between scatand path always exists, the angutering fields on the path larly fluctuation pattern in Fig. 6 follows the principle discussed in Section IV-A. 3) Fluctuation of Bistatic RCS in Weak-Coupling Region: As the target arises further and moves into the weak-coupling region in Fig. 5, the reflected waves of the underlying surface do not interfere with the incident wave. From (18), the . In the case of specular reflection, central axis is at the reflected wave is also a tapered wave with the central axis . When the target is located at , the is at horizontal distance between the target and the central axis of 2 , the amplitude of the the reflected wave is is thus as times reflected wave at . of the value at 50 Fig. 7 shows bistatic scattering when the target is at above rough sea surface. The amplitude of the specular reflected is so small as 0.0016 times of the value at . wave at almost cannot inTherefore, the reflected wave on the path terfere with the incident wave on the path . In Fig. 7, it can be seen that: 1) there are more frequent fluctuations of the bistatic RCS comparing with Fig. 6 and 2) these fluctuations are centered around the bistatic scattering pattern of the target in free space. Notice that in the direction of constructive interference, the bistatic RCS in Fig. 6 is nearly 12 dB above the bistatic RCS of the target in free space, i.e., the field intensity is 4 times higher. While in Fig. 7, this value is reduced to 6 dB, i.e., 2 times higher.
The reason is that in the strong-coupling region , both the and path , and illuminating wave interference on the path the scattered wave interference on the path and path exist, but in the weak-coupling region , only the latter is remained. The angular fluctuation of the bistatic RCS becomes smaller 90 , indicated by in Fig. 7. This near extreme LGA is due to insufficient length of the underlying sea surface in the numerical model: a fraction of the scattering wave on the path fails to be reflected by the underlying surface so that it cannot interfere with the scattered wave on the path . As the surface length is extended to 2000 in Fig. 8, it can be seen that the fluctuation of the bistatic RCS now becomes larger near the LGA, which agrees with the result of the image theory. It seems not necessary to remove this discrepancy because the surface length in numerical approach cannot be extended to infinity. In fact, due to the Earth curvature, the fluctuation reduction indicated by can be observed in practical measurement. C. Influence of the Target Altitude on Backscattering A relationship between backscattering coefficient and the target altitude is shown in Fig. 9, where the parameters of target, rough sea surface and incident wave are the same as Fig. 3. It can be seen that as the target altitude increases: 1) when 40 , backscattering RCS periodically fluctuates and gradually decays to show that scattering interaction between the target and 40 , the rough surface is reducing as increases; 2) when amplitude fluctuation keeps stable, because the target entered into the weak-coupling region ; and 3) the phase cancellation and path needs [11] of the path 2
1,2,3
(22)
where the odd and even correspond to the maximum and minimum of backscattering, respectively; (4) as increases and scattering interactions become weaker, the iteration number in our Robin-FFT method is reduced, and our method becomes
LIU AND JIN: AN FEM APPROACH WITH FFT ACCELERATED ITERATIVE ROBIN BOUNDARY
Fig. 9. Backscattering coefficient versus target altitude ( = 60 ).
Fig. 11.
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Backscattering coefficient versus target altitude ( = 80 ).
V. CONCLUSION
Fig. 10.
CPU time versus target altitude ( = 60 ).
more time efficient (see Fig. 10). The previous FEM-DDM approach has advantage in the case of low , but, as the altitude increases, the free space region quickly increased in the computation domain remarkably reduces its effectiveness. 4.5 , the maximum of In strong-coupling region backscattering RCS is nearly 12 dB above the case of target 40 , in free space. But in the weak-coupling region the difference is reduced to 6 dB. As mentioned early, it is due to absence of the illuminating wave interference in the weakcoupling region . Here the intensity of the specular reflected for 4.5 and 40 is 0.95 and 0.016 wave at , respectively. times of the value at As the incident angle is 80 in Fig. 11, comparing with Fig. 9, it can be seen that: 1) the fluctuation period of backscatin (22); 2) the fluctering RCS is extended as predicted by tuation decays faster, and the target enters the weak-coupling re13 , where the intensity of the specular gion as early as is 0.01 times of the value on the cenreflected wave at tral axis; 3) the fluctuation amplitudes almost keep stable when 40 , because the target is in the weak-coupling 13 region ; and 4) the fluctuation reduction is observed when is beyond 40 , which is due to insufficient length of the underlying surface in numerical approach at LGA as discussed in Figs. 7 and 8.
To study the scattering and interaction of a target arising from the rough oceanic surface, a FEM approach with FFT accelerated IRBC is developed. This approach presents an efficient simulation for the model of the target at high altitude above large-scale rough surface to show their strong or weak coupling along with the target altitude variation. Functional dependence of bistatic and back-scattering of the target and rough surface upon the target altitude, incident and scattering angle, etc., are numerically simulated and discussed. With improvement of the sea surface model and the adapt of domain decomposition method [12], the simulation can be further extended to larger scale rough surface and even higher altitude of the target.
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Peng Liu was born in Shanxi Province, China, in 1971. He received the B.S., M.S., and Ph.D. degrees from the Northwestern Polytechnical University (NWPU), Xi’an, China, in 1994, 1997, and 2001, respectively. From 2002 to 2004, he was a Postdoctoral Fellow at the Key Laboratory of Wave Scattering and Remote Sensing Information (Ministry of Education), Fudan University, Shanghai, China. He is currently an Associate Professor of the School of Information Science and Engineering, Fudan University. He has published over 20 papers in international and national journals. His main interests are EM theory and its application, computational electromagnetics, and remote sensing. Dr. Liu received the Excellent Student Award during his graduate study at NWPU.
Ya-Qiu Jin (SM’89–F04) received the B.S. degree from Peking University, Peking, China, in 1970, and the M.S., E.E., and Ph.D. degrees from the Massachusetts Institute of Technology, Cambridge, in 1982, 1983 and 1985, respectively. He was a Research Scientist at the Atmospheric and Environmental Research Inc., Cambridge, MA, in 1985, a Research Associate at the City University of New York from 1986 to 1987, a Visiting Professor, sponsored by the U.K. Royal Society, at the University of York, U.K., from 1993 to 1994, and at the City University of Hong Kong in 2001. He held the Senior Research Associateship at NOAA/NESDIS awarded by the USA National Research Council in 1996. He is currently a Professor in the School of Information Science and Engineering, and a Director of the Key Laboratory of Wave Scattering and Remote Sensing Information (Ministry of Education), Fudan University, Shanghai, China. He has been appointed as the Principal Scientist for the China State Major Basic Research Project for 2001 to 2006 by the Ministry of National Science and Technology of China. He has published over 440 papers in China and abroad, and eight books (three are in English: Electromagnetic Scattering Modeling for Quantitative Remote Sensing (Singapore: World Scientific,1994), Information of Electromagnetic Scattering and Radiative Transfer in Natural Media (Beijing: Science Press, 2000), and Theory and Approach of Information Retrieval from Electromagnetic Scattering and Remote Sensing (Germany: Springer, 2005). He is the Editor of SPIE Volume 3503: Microwave Remote Sensing of the Atmosphere and Environment (USA: SPIE), and the book: Wave Propagation, Scattering and Emission in Complex Media (Singapore: World Scientific and Beijing: Science Press, 2004). He is Chairman of ISAPE2000 and the Specialist Workshop 2004 on EM Scattering and Information Retrieval in Remote Sensing. His main research interests include scattering and radiative transfer in complex natural media, microwave remote sensing, including theoretical modeling, information retrieval and applications in atmosphere, ocean and earth surfaces, and computational electromagnetics. Dr. Jin received the China National Science Prize in 1993, the First-grade Science Prizes of the State Education Ministry in 1992 and 1996, the First-Grade State Guanghua Science Prize, the State Excellent Teacher Prize of Baosteel Foundation, the teaching excellence Prize of Shanghai City in 2001, the excellent supervisor for graduate students in Fudan University in 2003, and the Fudan President Prize in 2004, and appreciation for his notable service and contributions toward the advancement of IEEE professions from IEEE GRSS, among other many prizes. He was the Founder and Chairman of IEEE GRSS Beijing Chapter from 1998 to 2003.
