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Keywords:

  • domain decomposition method;
  • finite difference equation;
  • MEI;
  • cylinder scattering;
  • band matrix

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulations of the Finite Difference Equation
  5. 3. Domain Decomposition
  6. 4. Numerical Results
  7. 5. Conclusions
  8. References

[1] By combining the finite difference (FD) method with the domain decomposition method (DDM), a fast and rigorous algorithm is presented in this paper for the scattering analysis of extremely large objects. Unlike conventional methods, such as the method of moments (MOM) and FD method, etc., the new algorithm decomposes an original large domain into small subdomains and chooses the most efficient method to solve the electromagnetic (EM) equations on each subdomain individually. Therefore the computational complexity and scale are substantially reduced. The iterative procedure of the algorithm and the implementation of virtual boundary conditions are discussed in detail. During scattering analysis of an electrically large cylinder, the conformal band computational domain along the circumference of the cylinder is decomposed into sections, which results in a series of band matrices with very narrow band. Compared with the traditional FD method, it decreases the consumption of computer memory and CPU time from O(N2) to O(N/m) and O(N), respectively, where m is the number of subdomains and Nis the number of nodes or unknowns. Furthermore, this method can be easily applied for the analysis of arbitrary shaped cylinders because the subdomains can be divided into any possible form. On the other hand, increasing the number of subdomains will hardly increase the computing time, which makes it possible to analyze the EM scattering problems of extremely large cylinders only on a PC. The EM scattering by two-dimensional cylinders with maximum perimeter of 100,000 wavelengths is analyzed. Moreover, this method is very suitable for parallel computation, which can further promote the computational efficiency.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulations of the Finite Difference Equation
  5. 3. Domain Decomposition
  6. 4. Numerical Results
  7. 5. Conclusions
  8. References

[2] Recently, the EM scattering analysis of two-dimensional (2-D) objects has been playing an increasingly important role in electromagnetic field theory and practicable applications, especially in the radar cross-section (RCS) analysis [Arvas and Sarkar, 1989; Massoudi et al., 1988; Andreasen, 1964] and the wave propagation prediction in urban microcells of mobile communication networks [Xia et al., 1994; Kotterman et al., 1994; Schuster and Luebbers, 1997]. With the increase of frequency used in these applications a fast and rigorous method for solving the EM scattering by electrically large or even extremely large objects is needed.

[3] Many numerical methods have been proposed for solving such problems. As one of the most popular techniques, the method of moments (MOM) is widely used for the solution of integral equations defined on the surface of object [Arvas and Sarkar, 1989; Massoudi et al., 1988]. The finite difference (FD) method is another important method used for solving complex scattering problems of irregular, nonuniform objects [Mur, 1981]. Compared with MOM, the FD method results in a sparse coefficient matrix (with dimensions of Nf= L × Nc, where L is the number of grids in the normal direction and Nc is the number of grids along the cylinder circumference), whose consumption of computer memory and CPU time are O(N) and O(Nf2), respectively. However, the absorbing boundary condition (ABC) has to be applied to truncate the unbounded spatial domain. For most ABCs the distance between the cylinder surface and the truncated boundary must be large enough to reduce the spurious reflection on the truncated boundary. This demands a large L, which will lead to a huge coefficient matrix. Then, for large problems, this method will also exhaust computer resources.

[4] To promote the computational efficiency of the FD method, many works were focused on the improvement of ABCs, such as the development of the measured equation of invariance (MEI) and perfectly matched layer (PML) techniques. As a mesh termination boundary condition in the FD method, the MEI greatly reduced the dimensions of the FD matrix because the truncated boundary can be placed very close to the surface of the cylinder [Mei et al., 1994; Hong et al., 1994]. The fast-MEI further enhanced the computational efficiency of MEI [Zai and Hong, 1998]. By renumbering the nodes of the conformal meshes around the cylinder, a band matrix with very narrow bandwidth is then deduced in fast-MEI, whose computing cost and memory needs are both proportional to Nf. To convert the original sparse matrix into a narrow band matrix, however, it has to sacrifice nearly double the memory and is also not very efficient for some complex problems, such as the scattering analysis of concave cylinders. The PML method will encounter the same problem that the MEI method did.

[5] The domain decomposition method (DDM) is a promising method for solving large problems [Despr'es, 1992; Strupfel, 1996]. Unlike traditional methods, the DDM does not deal with the whole problem directly, but decomposes it into some independent small problems. For different subdomains, the most efficient method can be chosen independently for solving the EM equations on the subdomain. The DDM then greatly decreases the scale of the original problem. Moreover, it is well suited for numerical implementation on parallel computers and can considerably reduce both memory requirements and CPU time.

[6] In this paper, we present a novel fast algorithm based on the combination of the FD method and the DDM for electromagnetic scattering analysis of electrically large or extralarge conducting cylinders. The original FD method is introduced in section 2, and the implementation of the DDM under the FD equations is discussed in section 3. By decomposing the original domain into m subdomains along the cylinder circumference the new method obtains a series of band coefficient matrices with bandwidth (2N + 1)(Nis the number of mesh layers). This can both save memory and accelerate the solving process compared with fast-MEI. It can also be easily expanded to analyze some complex structures, for example, a cylinder with slots, by taking the slots as separated subdomains and treating them independently. Therefore the computing time and memory requirements of the new method are only proportional to NDand ND/M, respectively, and ND is the dimension of the coefficient matrix. These advantages make it possible to apply the new method to extremely large cylinders with complex cross sections. Some typical examples, including a circular cylinder, a rectangular cylinder, and a slotted cylinder with perimeter from 10λ0 to 100,000λ0 are computed and illustrated in section 4. All numerical results demonstrated the estimate of computational effort and memory requirements and agree very well with the data obtained by using MOM and MEI method.

2. Formulations of the Finite Difference Equation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulations of the Finite Difference Equation
  5. 3. Domain Decomposition
  6. 4. Numerical Results
  7. 5. Conclusions
  8. References

[7] The scattering problem of a 2-D perfectly conducting cylinder is shown in Figure 1. Conformal meshes are then created outside the surface as shown in Figure 2. Because of the axis of the cylinder being parallel to the zaxis, all field components are independent of variable z, and transverse magnetic (TM) or transverse electric (TE) polarized waves can independently exist in the system. Denoting the longitudinal component of the field by ϕ(x,y), the 2-D Helmholtz equation holds in the system:

  • equation image

The total field at each node consists of incident and scattered ones with the relation ϕt = ϕinc+ ϕs. The five-point difference equations can be easily derived by the loop integration technique [Kishk and Shafai, 1986] as

  • equation image

for interior nodes and

  • equation image

for the nodes on the layer just next to the surface of the cylinder. The expressions for the coefficients Ciare given by Zai and Hong[1998]. With the surface impedance boundary condition (SIBC)

  • equation image

where equation imageis the outgoing normal direction of the surface and the normalized impedance is

  • equation image

the FD equation for nodes on the surface of the cylinder can be written as b1ϕ1 + b2ϕ2= 0, where ϕ1 and ϕ2 are total fields on surface nodes and the nodes next to the surface, respectively. Here, b1 can be normalized as 1, and b2 = − exp (−jk0hνΔ) where hνis the normal mesh space. Separating the total field into the incident field and the scattered one yields

  • equation image

For the nodes on the truncated boundary, many ABCs can be employed on these nodes. As mentioned in section 1, boundary condition is not the emphasis of this paper because it is not the main factor for reducing the computational scale in this algorithm. Hence only two often-used ABCs of MEI and Mur condition of the second order are adopted in this paper. Both of them have the expression

  • equation image

By decomposing the Helmholtz operator (∇2 + k02) at the truncated boundary x = 0 the outgoing wave can be expressed as

  • equation image

expanding equation imagein series, and choosing the approximation of second order, we have

  • equation image

Approximating the partial derivatives with differences, (9) is reduced to (7), and the coefficients of the Mur condition of the second order are then determined.

image

Figure 1. EM scattering by 2-D conductor cylinder.

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image

Figure 2. Conformal FD meshes around the cylinder.

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[8] When applying MEI equations on the truncated boundary, the so-called MEI coefficients Cican be determined by “metrons,” which are defined as the fictitious currents on the surface of the cylinder

  • equation image

where L is the perimeter of the cylinder. The scattered fields at the MEI nodes produced by electric current Je(r′) or magnetic current Me(r′) can be obtained by the integrals

  • equation image
  • equation image

where riis the position vector of MEI nodes and r′ is the position vector on the surface of the cylinder. Here n′ is the outward normal direction of the surface. G(ri,r′) = − jH0[2](k0|rir′|)/4 is the Green's function of free space. For different metrons a system of linear equations with respect to the unknown MEI coefficients will result by substituting the corresponding scattering fields at the MEI nodes into (7). This system of linear equations can be solved easily.

[9] Combining the FD equations with MEIs (or Mur ABCs) together yields a matrix equation

  • equation image

where [S] is a sparse matrix, equation imageis the column matrix for unknowns (total fields ϕtat surface nodes and scattered fields ϕsat other nodes). Here, equation imageis a given column matrix introduced by the incident fields. The equivalent surface electric current is then derived from the total field near the surface of the cylinder:

  • equation image

3. Domain Decomposition

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulations of the Finite Difference Equation
  5. 3. Domain Decomposition
  6. 4. Numerical Results
  7. 5. Conclusions
  8. References

[10] It is known that the coefficient matrix [S] derived above is a sparse matrix, in which there are no more than five nonzero elements in each row. However, the inverse matrix of [S] will be a dense one and difficult to store economically. In addition, the inversion of [S] is hard to execute because it requires nearly O(N3) computational effort and the dimensions of [S] may be very large. Although some iterative procedures, such as the least squares method and generalized minimum residual (GMRES) method, etc., will not lead to fill-in, they still need O(Nα) (α ≈ 2) CPU time to approach the solution. By renumbering the nodes, fast-MEI can convert [S] to be a band matrix, which will reduce the computing time to O(N). Unfortunately, this method is restricted by the shape of the cylinder. For example, for a slot-embedded cylinder it is necessary to fill mesh in the slot. In this case, we cannot obtain a band matrix just by renumbering the nodes. Meanwhile, for some extremely large problems, the matrix [S] will be too huge to be operated directly.

[11] As shown in Figure 2, the conformal meshes around the cylinder form a loop. Let Ω be a boundary open set constructed by these meshes under the Helmholtz equation (equation (1)) and Γ = ∂Ω be the boundary of the region Ω. Now divide Ω into marbitrary overlapped subdomains along the circumference as illustrated in Figure 3. Let equation imagebe one of these subdomains, with equation image for contiguous Ωp and Ωq and with equation image for Ωp contiguous to Γ. Then the original problem defined by (1)is split into m small problems defined as follows:

  • equation image

where VBC is virtual boundary condition.

image

Figure 3. Domain decomposition along the circumference.

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[12] This kind of domain decomposition introduces three benefits: (1) Meshes in each subdomain will no longer form a loop, so a narrow band matrix with the bandwidth as (2Nl + 1) (Nl is the number of layers) will be obtained. This will much decrease the CPU time because the computation effort for such band matrices is only O(N) as mentioned in section 1. (2) The dimensions of [Si] derived in each subdomain will be reduced to 1/m of the original matrix [S] only. Therefore, for those electrically large cylinders, although the original matrix [S] is too huge to be stored and solved, the corresponding coefficient matrix for each subdomain can be small enough to be operated. This greatly reduces the requirements of computer resources and makes it possible to solve scattering problems by extremely large cylinders on a PC without restriction. Moreover, for those cylinders with complex shape, such as slotted cylinders, we can still decompose the meshes around the object and take the slot or other complex parts as separated subdomains, which is illustrated in Figure 4. These parts can be worked out independently with the most efficient method, which makes possible the flexibility of this algorithm and keeps the advantages mentioned above. (3) All these subdomains are independent from each other, so it is most suitable for parallel computation, which will further accelerate the whole algorithm.

image

Figure 4. Domain decomposition of a slotted cylinder.

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[13] Although the EM equation on each subdomain is solved individually, the global solution is obtained by an iterative procedure, and the virtual boundary condition on boundary Γq,p is adopted for exchanging data between subdomains. This procedure can be described in the following iterative form:

  • equation image

The construction of the virtual boundary condition is the key of the DDM because it majorly determines the convergence of the iterative procedure. In the traditional Schwarz alternating method the Dirichlet and Neumann conditions are adopted on Γq,palternately, but they are only valid for static field problems described with the Laplace equation. For dynamic field problems described with the Helmholtz equation, Despr'es [1992]introduced a new type of condition as

  • equation image

where νp is the outgoing normal of Ωp. Then the iterative algorithm (16) for solving Helmholtz equation (1)using DDM can be rewritten as

  • equation image

where ϕpnis the nth iterative solution in Ωp. It has been proven in Despr'es' [1992] work that the above iterative algorithm based on (17)has one and only one weak solution of (1).

[14] The VBC (equation (17)) should be converted to FD type when it is applied to the FD-DDM algorithm. Figure 5 shows two schemes for FD approximation of the VBC (equation (17)). If we adopt the scheme shown in Figure 5a, which means using the fields on node Dand node Dprev for difference approximation in subdomain Ωpwhile using the fields on node D and Dpostfor difference approximation in subdomain Ωq, it will be found that the iterative procedure does not converge. This is due to the error of O(h) in FD approximation. If we place the virtual boundary at the center between nodes rather than on the nodes, as shown in Figure 5b, and use the center difference to approximate the derivatives in VBC (equation (17)), the FD approximation error will be reduced to O(h2) and then the iterative procedure will converge. In this case, the FD equation corresponding to the VBC (equation (17)) is

  • equation image
image

Figure 5. Virtual boundary condition: (a) virtual boundary under FD grids and (b) improved virtual boundary under FD grids.

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4. Numerical Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulations of the Finite Difference Equation
  5. 3. Domain Decomposition
  6. 4. Numerical Results
  7. 5. Conclusions
  8. References

4.1. Convergence of the DDM

[15] Actually, Figure 5bpointed out that our algorithm is a kind of overlapped DDM, which means that the algorithm may have fast convergence. Figure 6demonstrates this conjecture and depicts the solutions of DDM with different iteration steps and compared with the solution found using MOM. From Figure 6 one can see that the results converge to the data derived from MOM quickly. In the meantime, the coefficient matrix [Si] for each subdomain is constant in each iteration step, they can be lower/upper triangular (LU) decomposed at the first step and then only need back-substitution in following steps. It is known that the computational time for back-substitution may be almost ignored, so the increase in the number of subdomains and iteration steps will hardly increase the CPU time. This conclusion is demonstrated in Figure 7. Hence we can save much memory nearly without increasing computing time.

image

Figure 6. Convergence of the new method.

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image

Figure 7. Relationship between the number of subdomains and computing time.

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4.2. Classical Scattering Problems

[16] The EM scattering by a 2-D cylinder with maximum perimeter of 100,000 wavelengths is computed and compared with the MOM and finite difference frequency domain (FDFD) method. Table 1 illustrates the comparison between these methods in both computational time and memory requirements. In Table 1 the equations derived by the MOM are solved by LU decomposition, and the equations derived by FDFD method are solved by least squares technique. Mur's condition of the second order is used both in FDFD method and the DDM, and there are 10 layers of meshes between the truncated boundary and the surface of the cylinder. All these methods have the same mesh spaces, namely, 10 mesh nodes per wavelength. For DDM the maximum number of subdomains is chosen as 20 in the computation. Table 2lists the comparison between fast-MEI and DDM. MEIs were used for truncating the meshes in both of these two methods. Although they have almost the same computational effort (in fact, DDM is slightly faster because the bandwidth of matrix introduced by it is nearly half of that derived by fast-MEI), DDM can save much computer memory compared with the fast-MEI method. The memory consumption and computing time for different sizes of scatterer are also plotted in Figure 8. These comparisons show substantial computer resource saving, which also verified the conclusions predicted theoretically. Figure 9 illustrates the surface currents of a concave cylinder calculated by DDM and MOM, and the results are in good agreement. In this case, we still decompose the domain along the circumference and take the mesh in the cavity as a single subdomain. So all subordinate coefficient matrices can still be treated as band matrices. It costs 0.51 s by the DDM, while it costs 28.67 s by MOM. Figures 10 and 11 show the surface currents of a rectangular cylinder with perimeter of 10,000 wavelengths and a circular cylinder with perimeter of 100,000 wavelengths. These two test cases are used to demonstrate the advantage of our algorithm in solving extralarge problems. By partitioning the huge matrix into a series of small matrices for each subdomain, we can easily work them out one by one with affordable memory and CPU time cost. Certainly, other methods, such as the geometrical theory of diffraction or uniform theory of diffraction, or even MOM (by applying some special treatment, e.g., the quasicomplete domain basis function), can yield these results too. On the other hand, however, these methods may not handle complex shapes or may consume more memory compared with this new algorithm. All the numerical results demonstrated the validity of the new method.

image

Figure 8. Comparison between fast-MEI and DDM in computing time and memory.

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image

Figure 9. Surface current of a concave cylinder.

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image

Figure 10. Surface current of a rectangular cylinder with perimeter of 10,000 wavelengths.

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image

Figure 11. Surface current of a circular cylinder with perimeter of 100,000 wavelengths.

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5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulations of the Finite Difference Equation
  5. 3. Domain Decomposition
  6. 4. Numerical Results
  7. 5. Conclusions
  8. References

[17] The formulations and numerical results presented in section 4 illustrate that DDM is a fast and rigorous method for solving EM scattering problems of electrically large objects. By using this method one need not face the whole problem directly but independent small problems. The decomposition results in a series of band matrices with bandwidth (2Nl + 1) for each subdomain, which decreases the computing time and memory consumption from O(N2) for the traditional FDFD method to O(N) and O(N/m), respectively. As mentioned in section 4.1, the increase in the number of subdomains will hardly increase the computing time. On the other hand, these subdomains can be divided in any possible form, so it provides a possibility for analyzing the scattering problems of arbitrarily shaped electrically large and complex objects. Numerical results show that the new DDM-based method is reliable, efficient, and convenient. Moreover, each subdomain can be processed independently, so it is very suitable for parallel computation, which can accelerate the algorithm further.

Table 1. Comparison Among Different Methods in Computation Time and Memory Requirementsa
Perimeter λ0MOMFDFDDDM
Time, sMemory. MbTime, sMemory, MbTime, sMemory, Mb
  • a

    Machine type: PII-300, 256 Mb RAM.

125.163.392.770.780.11<0.7
2014.975.074.930.920.15<0.7
3052.437.658.091.100.19<0.7
40116.6710.012.321.290.24<0.7
50215.1414.015.761.470.51<0.7
80  23.292.020.610.93
100  38.502.320.721.26
500  117.889.603.061.37
800  191.6517.684.831.81
1000  331.0022.006.252.10
2000  542.1845.0010.913.55
10,000  2439.81230.0030.2014.00
40,000    251.7057.00
100,000    626.30141.00
Table 2. Comparison Between the Fast-MEI Method and DDMa
Perimeter λ0Fast-MEIDDM
Time, sMemory, MbTime, sMemory, MbNumber of Subdomains
  • a

    Machine type: PII-400, 512 Mb RAM.

10006.045.144.83.7910
200011.089.219.76.8610
400021.0177.0019.112.8710
800043.0353.0039.119.3615
10,00055.0441.0048.120.4920

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Formulations of the Finite Difference Equation
  5. 3. Domain Decomposition
  6. 4. Numerical Results
  7. 5. Conclusions
  8. References
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  • Arvas, E., and T. K. Sarkar, RCS of two-dimensional structures consisting of both dielectric and conductors of arbitrary cross section, IEEE Trans. Antennas Propag., 37, 546554, 1989.
  • Despr'es, B., A domain decomposition method for the harmonic Maxwell equations, in Iterative Methods in Linear Algebra, pp. 475484, Elsevier Sci. New York, 1992.
  • Hong, W., Y. W. Liu, and K. K. Mei, Application of the measured equation of invariance to solve scattering problems involving penetrable medium, Radio Sci., 29(4), 897906, 1994.
  • Kishk, A. A., and L. Shafai, Different formulations for numerical solution of single and multibodies of revolution with mixed boundary conditions, IEEE Trans. Antennas Propag., 34, 666673, 1986.
  • Kotterman, W. A. T., A. Mawira, and G. J. C. Verhoef, Investigation on the influence of large buildings on the propagation of radio waves, paper presented at 5th IEEE International Symposium on Personal, Indoor and Mobile Radio Communication (PIMRC'94), Inst. of Electr. and Electron. Eng. The Hague, Netherlands, 18–22 Sept. 1994.
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  • Mei, K. K., R. Pous, Z. Chen, Y. W. Liu, and M. D. Prouty, Measured equation of invariance: A new concept in field computation, IEEE Trans. Antennas Propag., 37, 320328, 1994.
  • Mur, G., Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations, IEEE Trans. Electromagn. Compat., 23, 377382, 1981.
  • Schuster, J. W., and R. J. Luebbers, Comparison of GTD and FDTD predictions for UHF radio wave propagation in a simple outdoor urban environment, paper presented at IEEE Antennas and Propagation Symposium, Inst. of Electr. and Electron. Eng. Montreal, Que., Canada, July 1997.
  • Strupfel, B., A fast-domain decomposition method for the solution of electromagnetic scattering by large objects, IEEE Trans. Antennas Propag., 44, 13751385, 1996.
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