Efficient analysis of coated object using fast algorithm with surface impedance boundary condition

Abstract

An effective solution method is proposed to analyze the electromagnetic scattering problem of coated targets, which is based on a novel form of multilevel matrix decomposition algorithm (NFMLMDA). It uses an efficient directional grouping scheme to subdivide the far-field domain, and then a novel distribution form of equivalent source is applied to reduce the matrix filling time and memory requirement. By using the grouping scheme, the far-field interaction domain can be divided into many cone structures. The matrix between the observation group and far-field group in the cone structure is low-rank, which meets the directional far-field requirement. Meanwhile, the impedance boundary condition (IBC) is applied to simplify the solution process and improve the accuracy of the approximate solution. The method of this paper is very suitable for the needs of various kinds of complex problems.

Keywords

Novel form of multilevel matrix decomposition algorithm (NFMLMDA)fast solution impedance boundary condition (IBC)

1. Introduction

The development of modern radar detection technology has put forward strict requirements for the low radar cross section (RCS) characteristics of aircraft. In addition to stealth design, the coating radar absorbing materials on the surface of aircraft is one of the most basic means to reduce radar cross section (RCS). Because the aircraft has almost stringent requirements on the load of its components, the coated radar absorbing materials must have the characteristics of light weight and thin thickness, that is to say, the minimum coated materials are required to obtain the maximum RCS reduction effect. In order to evaluate the performance of absorbing materials and design the coating scheme, electromagnetic simulation modeling and numerical analysis are often faster and more economical than experimental testing, and are not affected by the external testing environment.

There are many analysis schemes for electromagnetic scattering modeling of coated structures of absorbing materials. The method of moments (MoM) [1,2] is a popular method for solution of electromagnetic problems, which is more suitable for the analysis of open domain problem than finite difference time domain (FDTD) [3] and finite element method (FEM) [4–6], is the main method to solve the problems of the coated radar absorbing materials. However, the coating thickness of radar absorbing materials used for stealth is often very thin, and the electromagnetic parameters of dielectric absorbing materials are often very large. All these bring difficulties to geometric modeling and mesh generation of surface integral equation (SIE) and volume-surface integral equation (VSIE), which make the mesh distribution locally dense and make the condition number of impedance matrix worse. The calculation accuracy and efficiency of electromagnetic simulation calculation are seriously affected. The impedance boundary condition (IBC) [7,8] electromagnetic model based on integral equation method is often used to analyze electromagnetic scattering of the thin coated structures.

To improve the efficiency of the whole algorithm, all operations, including that used to obtain the sparse representation of the full matrix from the mathematical representation, should only operate on the sparse representations of the system matrix. Some available techniques are applied to obtain a sparse matrix representation from mathematical formulations, include the adaptive cross approximation (ACA) [9–12], multilevel matrix decomposition algorithm (MLMDA) [13,14], Multilevel UV method [15] and H-matrix methods [16,17]. In this paper, a novel nested form of MLMDA algorithm is applied to fill the impedance matrix efficiently, which utilizes the directional grouping scheme to subdivide the far-field domain. Fast iterative methods provide efficient strategies for calculating the matrix-vector multiply, which save both CPU time and memory. However, these methods often suffer from a lack of robustness. Meanwhile, the impedance boundary condition (IBC) is applied to simplify the solution process and improve the accuracy of the approximate solution. The proposed method is very suitable for various kinds of complex problems.

This paper is organized as follows. Section 2 describes the proposed method in more detail. Section 3 gives some numerical examples to demonstrate the accuracy and computation efficiency of our approach.

2. Theory

2.1. Impedance boundary condition (IBC)

According to the impedance boundary conditions, the electric and magnetic fields of the coated target surface are satisfied, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \overline{n}\times (\overline{n}\times E)=-{\eta}_{0}{\eta}_{s}(\overline{n}\times H)=-\left.E\right|_{\tan }\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \overline{n}\times (\overline{n}\times H)=-{\eta}_{0}{\eta}_{s}(\overline{n}\times E)=-\left.H\right|_{\tan }\end{eqnarray}$ (2) where 𝜂₀ is the free space wave impedance, n is the external normal vector of the coating surface. The 𝜂_s denotes the relative surface impedance, which can be expressed as, $\begin{eqnarray}\displaystyle {\eta}_{s}=-i\sqrt{{\mu}_{r}/{\varepsilon}_{r}}\tan (\sqrt{{\mu}_{r}{\varepsilon}_{r}}k_{0}d). & & \displaystyle\end{eqnarray}$ (3) Where ϵ_r and μ_r represent the relative permittivity and relative permeability of the medium respectively. k ₀ denotes the wave number of electromagnetic wave in free space and d denotes the coating thickness.

According to the equivalent principle, the relationship between surface current J and magnetic current M can be expressed as, $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle J={\displaystyle \frac{1}{{\eta}_{0}{\eta}_{s}}}\overline{n}\times M\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle M=-{\eta}_{0}{\eta}_{s}\overline{n}\times J.\end{eqnarray}$ (5)

By substituting E = E ^inc + E ^s, H = H ^inc + H ^s and ((5)) into ((2)) and ((3)), the integral equations can be written as, $\begin{eqnarray}\displaystyle [{\eta}_{0}L(J)+{\eta}_{0}{\eta}_{s}K(\overline{n}^{\prime }\times J)]_{\tan }-{\eta}_{0}{\eta}_{s}\overline{n}\times [K(J)-{\eta}_{s}L(\overline{n}^{\prime }\times J)]={\eta}_{0}{\eta}_{s}\overline{n}\times H^{inc}-\left.E^{inc}\right|_{\tan } & & \displaystyle\end{eqnarray}$ (6) where, $\begin{eqnarray} \displaystyle \text{} & \text{} & \displaystyle L(X)=ik\int _{s}\bigg[X(r^{\prime })+{\displaystyle \frac{1}{k_{0}^{2}}}{\nabla}{\nabla}^{^{\prime }}\cdot X(r^{\prime })\bigg]G(r,r^{\prime })dS^{\prime } \end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K(X)=P.V.\int _{s}{\nabla}G(r,r^{\prime })\times X(r^{\prime })dS^{\prime }\end{eqnarray}$ (8) where P. V. represents the main value integral, G (r, r ^′) denotes Green’s function in free space. The surface current is expanded to RWG basis function, and the matrix equation can be established by using the Galerkin matching method.

2.2. Directional grouping scheme

In this paper, an efficient directional grouping scheme is applied to form a new tree structure to divide the far-field and near-field information. The main idea of directional grouping scheme [18–20] is similar to the low-rank decomposition technique, which divides the far-field domain into many wedges. It satisfies the directional parabolic separation condition. The interactions between observation group and the boxes in the wedge can be accelerated using the low-rank representation.

The object is firstly divided into many sub-domains by tree structure, which is shown in Fig. 1.

Fig. 1.

Subdivision of an object using bounding boxes.

Then, the computation is organized in a multi-directional way. For a box Y of size w, its far field, defined to be the region at least w ² away from Y, is partitioned into a group of wedges that satisfy the directional parabolic separation condition (see Fig. 2(a)). The calculation of the interactions between Y and all of the boxes in a specific wedge can be accelerated using the low-rank representation associated with this wedge. This framework is repeated recursively at all levels (see Fig. 2(b)).

Fig. 2.

(a) The directional grouping scheme in the three dimensional case. (b) The multilevel directional grouping scheme.

2.3. Equivalent source

The principle of multilevel matrix decomposition algorithm (MLMDA) is based on Huygens’ equivalence principle, which mainly uses the equivalent RWG [21] to calculate the far-field region quickly. The Huygens’ equivalence principle states that the scattered field from a scatterer can be replaced by the field radiated by equivalent sources on a surface S which encloses the scatterer. The distribution form of a wedge is shown in Fig. 3.

Fig. 3.

The distribution form of equivalent sources in the three dimensional case.

When the object is divided into multilevel form, the equivalent sources of finer level is used as the unknown to calculate matrix information of parent’s level.

2.4. Novel form of multilevel matrix decomposition algorithm (NFMLMDA)

When the equivalent sources are set, a novel form of multilevel matrix decomposition algorithm is proposed to accelerate the matrix vector multiplication. The multilevel matrix decomposition algorithm (MLMDA) is first proposed in [13], which evaluates the multiplication of the submatrices with a trial solution vector using a multilevel algorithm that resembles an FFT. It can reduces the memory requirements and the computational complexity to O(Nlog²N). In this paper, a novel expression form of MLMDA is introduced. The specific expression is expressed in the following,

(1) There are n _l and m _l original RWG functions in the source and observation boxes at level l, respectively. Similarly, p _l and q _l, respectively, refer to the indexes of the equivalent RWG functions at the boundary of these source and observation boxes. For a given observation box Y, the interaction between the original RWG functions of Y and the equivalent RWG functions of a wedge can be expressed as Z _{n
_l
q
_l}. Then, a truncated SVD is applied to decompose Z _{n
_l
q
_l}into, $\begin{eqnarray}\displaystyle Z_{n_{l}q_{l}}=Z_{n_{l}r}\cdot S_{rr}\cdot Z_{rq_{l}} & & \displaystyle\end{eqnarray}$ (9) where r denotes the rank of Z _{n
_l
q
_l}, which is smaller than q _l and n _l. It requires O (r (n _l + r + q _l)) ≪ O (n _l q _l) to store Z _{n
_l
r} ⋅ S _rr ⋅ Z _{rq
_l}.

(2) The interaction between the equivalent RWG functions of Y and the original RWG functions of a wedge can be expressed as Z _{p
_l
m
_l}. Then, a truncated SVD is applied to decompose Z _{p
_l
m
_l} into, $\begin{eqnarray}\displaystyle Z_{p_{l}m_{l}}=Z_{p_{l}r^{\prime }}\cdot S_{r^{\prime }r^{\prime }}\cdot Z_{r^{\prime }m_{l}} & & \displaystyle\end{eqnarray}$ (10) where r ^′ denotes the rank of Z _{p
_l
m
_l}, which is smaller than p _l and m _l. As similar to the previous result, it requires O (r ^′(p _l + r ^′ + m _l)) ≪ O (p _l m _l) to store Z _{p
_l
r
^′}⋅ S _{r
^′
r
^′}⋅ Z _{r
^′
m
_l}.

(3) The interaction between the equivalent RWG functions of Y and the equivalent RWG functions of a wedge can be expressed as Z _{p
_l
q
_l}.

Fig. 4.

(a) The three-dimension of the ogive-sphere structure with homogeneous coating; (b) The two-dimension profiles of the ogive-sphere structure with homogeneous coating.

Fig. 5.

The bistatic scattering cross sections of an ogive-sphere configuration with homogeneous coating.

Fig. 6.

The memory requirements of NLMLMDA for the coated ogive-sphere.

Fig. 7.

The computational complexity of NLMLMDA for the coated ogive-sphere.

Through the above steps, the interaction between the original RWG functions of Y and the original RWG functions of a wedge can be expressed as, $\begin{eqnarray}\displaystyle Z_{n_{l}m_{l}}\text{} & =\text{} & \displaystyle Z_{n_{l}q_{l}}\cdot Z_{p_{l}q_{l}}^{-1}\cdot Z_{p_{l}m_{l}}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle Z_{n_{l}r}\cdot S_{rr}\cdot Z_{rq_{l}}\cdot Z_{p_{l}q_{l}}^{-1}\cdot Z_{p_{l}r^{\prime }}\cdot S_{r^{\prime }r^{\prime }}\cdot Z_{r^{\prime }m_{l}}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle Z_{n_{l}r}\cdot S_{rr^{\prime }}^{\prime }\cdot Z_{r^{\prime }m_{l}}\end{eqnarray}$ (11) where the dimension of S ^′ is very small.

When the object is divided into multilevel form, the p _l and q _l of finer level’s equivalent RWG functions become the source RWG of the coarser level. The far-field matrix of level l-1 can be expressed as, $\begin{eqnarray}\displaystyle Z_{l-1}^{far}=Z_{n_{l}q_{l}}\cdot Z_{q_{l}p_{l-1}}\cdot Z_{q_{l-1}p_{l-1}}^{-1}\cdot Z_{q_{l-1}p_{l}}\cdot Z_{p_{l}m_{l}}. & & \displaystyle\end{eqnarray}$ (12)

3. Numerical results

In this section, some numerical results are presented to demonstrate the efficiency of the proposed method in solving linear systems of electromagnetic scattering problems.

The simulation results of an coated ogive-sphere are given to demonstrate the efficiency of the novel form of multilevel matrix decomposition algorithm (NFMLMDA). The ogive-sphere shown in Fig. 4 is covered with a uniform coating with ϵ_r = 99.75 − j10.0, μ_r = 10. The specific dimension parameters of the coated ogive-sphere are shown in Fig. 4(b).

The bistatic scattering cross sections of the coated ogive-sphere structure is given in Fig. 5. The direct filling means that the whole impedance matrix is filled by the method of moments. It can be observed that the results of NFMLMDA is agree well with that of direct filling. Meanwhile, the matrix-vector multiplication time and the memory requirement of NFMLMDA are given in Figs 6–7. It can be found that NFMLMDA is much more efficient than traditional MLACA and MLMDA. Meanwhile, the MVP time is less than that of MLFMA [22,23]. It can reduces the memory requirements and the computational complexity to O(Nlog²N).

4. Conclusion

In this paper, a novel nested form of multilevel matrix decomposition algorithm (NFMLMDA) is introduced to efficiently analyze the scattering of coated structures. It utilizes an efficient directional grouping scheme to subdivide the far-field domain, and then a novel equivalent source distribution form is applied to reduce the matrix filling time and memory requirement. Compared with the conventional decomposition method, the NFMLMDA method can reduce the computational time significantly for the scattering of coated structures.

References

Essid

Bassem Ben Salah

and Samet

, Hybrid spatial-spectral MoM analysis of microstrip structures, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 29(4) (2016), 763–772.

Imre

Tamás

B.P.

and Szabolcs

, Fast analysis of metallic antennas by parallel moment method implemented on CUDA, International Journal of Applied Electromagnetics and Mechanics 39 (2012), 677–683.

Chaudhury

Chattopadhyay

P.K.

Raju

and Chaturvedi

, Transient analysis of creeping wave modes using 3-D FDTD simulation and SVD method, IEEE Trans. on Antennas and Propagation 57(3) 2009.

Wan

X. Q.

and Chen

R. S.

, Hierarchical LU decomposition based direct method with improved solution for 3D scattering problems in FEM, Microw. Opt. Technol. Lett. 53(8) (2011), 1687–1694.

Zhang

H.H.

Lin

Sha

W.E.I.

Choi

W.W.

Tam

K.W.

Garcia-Donoro

and Shi

, Electromagnetic-thermal analysis of human head exposed to cell phones with the consideration of radiative cooling, IEEE Antennas and Wireless Propagation Letters 17(9) (2018), 1584–1587.

Gan

H.H.

Xia

Dai

Q.I.

and Chew

, Augmented electric field integral equation for inhomogeneous media, IEEE Antenn. Wireless Propag. Lett. 16 (2017), 2967–2970.

Medgyesi-Mitschang

L.N.

and Wang

D.S.

, Hybrid methods for analysis of complex scatterers, Proceedings of the IEEE 77(5) 1989.

Ozdemir

Akduman

Yapar

and Crocco

, Higher order inhomogeneous impedance boundary conditions for perfectly conducting objects, IEEE Transactions on Geoscience and Remote Sensing 45(5) (2007).

Kezhong

, The adaptive cross approximation algorithm for accelerated method of moments computations of EMC problems, IEEE Transactions on Electromagnetic Compatibility 47 (2005), 763–773.

10.

Hislop

, Efficient sampling of electromagnetic fields via the adaptive cross approximation, IEEE Transactions on Antennas and Propagation 55 (2007), 3721–3725.

11.

Bebendorf

, Approximation of boundary element matrices, Numerische Mathematik 86 (2000), 565–589.

12.

Zhang

H.H.

Wang

Q.Q.

Shi

Y.F.

and Chen

R.S.

, Efficient marching-on-in-degree solver of time domain integral equation with adaptive cross approximation algorithm-singular value decomposition, Appl. Comput. Electromagn. Soc. J. 27(6) (2012), 475–482.

13.

Michielssen

and Boag

, A multilevel matrix decomposition algorithm for analyzing scattering from large structures, IEEE Trans. Antennas Propag 44(8) (1996), 1086–1093.

14.

Heldring

Tamayo

J.M.

Rius

J.M.

and Parron

, Multilevel MDA-CBI for fast direct solution of large scattering and radiation problems, IEEE Antennas and Propagation Society, AP-S International Symposium (Digest) (2007), 5599–5602.

15.

Deng

F.S.

S.Y.

Chen

H.T.

W.D.

W.X.

and Zhu

G.Q.

, Numerical simulation of vector wave scattering from the target and rough surface composite model With 3-D multilevel UV method, IEEE Trans. Antennas Propag. 58(5) (2010), 1625–1634.

16.

Liu

and Jiao

, Existence of H-matrix representations of the inverse finite-element matrix of electro-dynamic problems and H-based fast direct finite-element solvers, IEEE Trans. Microw. Theory Tech. 58(12) (2010), 3697–3709.

17.

Bebendorf

and Hackbusch

, Existence of H-matrix approximants to the inverse FE-matrix of elliptic operators with L-infinity-coefficients, Numerische Mathematik. 95(1) (2003), 1–28.

18.

Engquist

and Ying

, Fast directional multilevel algorithms for oscillatory kernels, SIAM Journal on Scientific Computing 29(4) (2007), 1710–1737.

19.

Goreinov

S.A.

Tyrtyshnikov

E.E.

and Zamarashkin

N.L.

, A theory of pseudoskeleton approximations, Linear Algebra Appl 261 (1997), 1–21.

20.

Candès

Demanet

and Ying

, Fast computation of fourier integral operators, SIAM Journal on Scientific Computing 29(6) (2007), 2464–2493.

21.

Rius

J.M.

Parron

Ubeda

and Mosig

, Multilevel matrix decomposition algorithm for analysis of electrically large electromagnetic problems in 3-D, Microw Opt. Technol. Lett 22(3) (1999), 177–182.

22.

Dai

Q.I.

Gan

Liu

Chew

and Wei

, Large-scale characteristic mode analysis with fast multipole algorithms, IEEE Trans. Antennas Propagat. 64 (2016), 2608–2616.

23.

Xia

Meng

Liu

Q.S.

Gan

H.H.

and Chew

, A low-frequency stable broadband multilevel fast multipole algorithm using plane wave multipole hybridization, IEEE Trans. Antennas Propagat. 66(11) (2018), 6137–6145.