An Overlapped Domain Decomposition Method for Solving Electromagnetic Surface Integral Equations Jun Hu, Wei Hong, Hou-Xing Zhou, Wei-Dong Li, Zhe Song State Key Laboratory of Millimeter Waves School of Information Science and Engineering, Southeast University Nanjing, 210096, P. R. China E-Mail:
[email protected];
[email protected] Abstract—An overlapped domain decomposition method with congruent sub-domains is developed for electromagnetic surface integral equations. Based on congruent sub-domains, the proposed method creates a set of uniform mesh such that all the self-matrices on the congruent sub-domains share an interiorinterior sub-matrix, which can reduce the memory requirement and the matrix filling time, intensively, especially for extremely large objects with great numbers of congruent sub-domains.
the IE-ODDM. The results of several examples in this paper show the good performance of the extended IE-CSD-ODDM. II. THE PRINCIPLE OF THE IE-CSD-ODDM In the MoM, the numerical solution of the surface integral equations (SIE) results in solving a linear equation ZI V , (1) where Z is the coefficient matrix, I is the unknown coefficient vector of the current (simply called the current vector), and V is the excitation vector of the incident field. The IE-ODDM divides the whole large dense matrix Z into many smaller sub-matrices according to a given domain decomposition scheme such that each sub-matrix equation can be easily solved. Applying the IE-ODDM [8], one can reduce Equation (1) to the following form
I. INTRODUCTION The method of moments (MoM) has been widely used to solve electromagnetic (EM) scattering problems [1]. Compared with some other methods, the number of unknowns of the MoM is relatively small. However, the dense matrix was a bottleneck in the MoM application until the fast multipole method [2], the multi-level fast multipole algorithm (MLFMA) [3, 4], the adaptive integral method [5], and the domain decomposition methods, including the forward backward iteration scheme [6, 7] and the overlapped domain decomposition method (IE-ODDM) [8], were proposed. Recently, the IE-ODDM-MLFMA, a hybrid method, has been built [9, 10], which utilizes the MLFMA for enhancing the computational efficiency of the IE-ODDM, and has been reported by many researchers [11-15]. In the practice of the IE-ODDM, we find some interesting phenomena. Some objects have surfaces consisting of congruent geometries; some other objects have surfaces with large sub-domains which can be approximated by congruent geometries [16]. The above domain decomposition will make the MoM self-matrices on each congruent sub-domain almost identical. In [16], an overlapped domain decomposition method with congruent sub-domain (IE-CSD-ODDM) is developed, which tries to utilize this character to enhance the efficiency of the IE-ODDM. That method suggests a good expectation that IE-CSD-ODDM can reduce the memory requirement and the filling time of the matrix, intensively, even if it is limited to the objects with fully congruent triangle sub-domains. In this paper, the IE-CSD-ODDM is extended to more general objects with several types of congruent subdomains, and the congruent sub-domains are not only the triangle-type sub-domains but also the curve-surface-type subdomains. In the present method, the characters of the geometry of objects, such as congruent sub-domains and symmetrical sub-domains, can be made full use of to enhance
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§ Z A( TI lA)I ¨ (T ) ¨ ZA l A © E ,B I where
Z A(TI lA)E , B · § I A( Tl ) · ¸ ¨ ( TI ) ¸ Z A( TEl, B) AE , B ¸¹ ¨© I AEl, B ¸¹
VA(ITl ) ° ® (Tl ) °¯VAE , B
§ VA(ITl ) · ¨¨ (Tl ) ¸¸ , © VAE , B ¹
VA(ITl ) Z A( TlA)C I A(TCl ) I
V
( Tl ) AE , B
Z
( Tl ) AE , B A C
I A(TCl )
.
(2)
(3)
The explanations of the above subscript can found in [16]. Equation (2) represents an EM scattering problem on the sub(T )
domain Tl , and the exterior current I A Cl
in (3) can be
regarded as the additional incident source. The coefficient matrix on the left hand side of (2) is called the self-matrix in this paper. After Equation (2) is solved, only the current on Tl is used to update the entire current vector I on the whole domain. The above procedure should be repeated till the relative residual error is less than the given threshold. For Equation (2), there are many solving methods, such as the direct method and the iteration method. In [16], a modified direct method is efficiently used to solve Equation (2) on the congruent triangle sub-domains, which can utilize the congruent characteristic of sub-domains and avoid the repetition of the iteration method for hundreds of right hand terms. For a large surface domain with many congruent subdomains, one can choose a uniform meshing scheme such that
ICCP 2010 Proceedings
TABLE II MATRIX FILLING TIME OF THE FOUR EXAMPLES
(T )
their interior-interior sub-matrices Z AI lAI become consistent with each other. Considering that the interior-interior submatrix occupies most of the space in the self-matrix, the total storage for all the self-matrices will be significantly reduced by sharing a common interior-interior sub-matrix. Moreover, the LU decomposition of the common sub-matrix is calculated and stored in advance to enhance the efficiency. In [16], the solution of Equation (2) on the congruent subdomains can be explicitly written as
1
I ( Tl ) C ( Tl ) L1Z A(TI lA)E , B Z A(TEl, )B AE , B ° AE , B ° . (3) C ( Tl ) L1VA(ITl ) VA(ET,lB) ® ° (T ) °UI AI l L1VA(ITl ) L1Z A( TI lA)E , B I A(TEl, )B ¯ where L and U are the LU decomposition matrices of the (T ) (T ) common matrix Z AI lAI , and the transforming matrix C l is
Example
I
II
III
IV
IE-CSD -ODDM
1.977 m
1.110 m
18.63 m
2.006 m
IE-ODDM
2.603 m
1.509 m
26.49 m
2.280 m
Filling time reduction (%)
24.05
26.44
29.67
12.08
of the interior-interior sub-matrix of each type sub-domain is calculated and stored; 7) Building the formulae (3) for each congruent sub-domain and solving Equation (2) for currents on each of sub-domains till the convergent result is obtained. Not that Equation (2) on each of congruent sub-domain is solved by (3), and on the other un-congruent sub-domain can be solved by other usual methods.
referred to [16]. Equation (3) is that modified direct solution of Equation (2) on the congruent domain. The above description shows the principle of the IE-CSDODDM for congruent triangle sub-domains. More details can be found in [16]. The basis of the IE-CSD-ODDM is the congruent sub-domain decomposition scheme and the uniform meshing. If the sub-domains are congruent triangle subdomains, the uniform meshing can be easily obtained by the sub-division, however, it is not the same thing for arbitrary sub-domains. Some more skills must be employed. The following outline shows the uniforming meshing procedure and the whole IE-CSD-ODDM for complex objects with several types of congruent sub-domains. 1) Meshing all the sub-domains to get its triangle patches respectively. Note that only one sub-domain of each type of congruent sub-domains is meshed; 2) Transforming the mesh obtained on the selected subdomains of each type domain to all the other sub-domains of the same type; 3) Combining all the local triangle meshes on sub-domains for a global mesh on the whole domain; 4) Generating all the RWG bases on the whole domain; 5) Rearranging the local indexes of the RWGs of the congruent sub-domains so that the self-matrices of each type congruent sub-domain can have an identical interior-interior sub-matrix; 6) Generating all the self-matrices. Note that only one copy
III. NUMERICAL EXAMPLES In order to validate the proposed method, four kinds of typical PEC objects, i.e., cubes, sphere, missile-type object and airplane-type object, are considered. The incident EM field is a plane wave with the x -axis polarization and the z axis propagation direction. The program runs on a workstation with two 2.50 GHz CPUs and utilizes seven threads synchronous. Example I: A PEC cube is considered, which is paved with 12 congruent triangle sub-domains, as shown in Fig. 3. The object has the edge lengths 2O and involves 7200 unknowns. The results obtained by the IE-ODDM and the IE-CSDODDM are compared, as shown in Fig. 3. The result by the IE-CSD-ODDM is well consistent with that by the IE-ODDM. Table I lists the memory requirements of self-matrices in the IE-ODDM and the IE-CSD-ODDM, which shows that the IECSD-ODDM can reduce about 29.74% from the memory requirement in the IE-ODDM. Corresponding to Table I, Table II lists the filling time of the self-matrices in the two different methods, which shows that the IE-CSD-ODDM can reduce about 24.05% from the filling time in the IE-ODDM. Example II: A PEC sphere is considered, which consists of 8 congruent sub-domains, as shown in Fig. 4. The object has the radius O and involves 4584 unknowns. The results by the two
TABLE I MEMORY REQUIREMENTS OF THE FOUR EXAMPLES
Example
I
II
III
IV
IE-CSD -ODDM
130.4 MB
65.15 MB
891.7 MB
118.4 MB
IE-ODDM
185.6 MB
97.34 MB
1.712 GB
143.0 MB
Memory reduction (%)
29.74
30.94
50.86
17.20
TABLE III ITERATION TIME OF THE FOUR EXAMPLES
Example
I
II
III
IV
IE-CSD -ODDM
81.84 s
7.891 s
4.479 m
27.00 s
IE-ODDM
74.60 s
9.113 s
4.980 m
28.26 s
TABLE IV PRIMARY PARAMETERS OF THE FOUR OBJECTS
Example
I
II
Object
Cube
Sphere
12
Number of sub-domains Number of types Number of unknowns
III Missiletype
IV Airplanetype
8
13
22
1
1
3
6
7200
4584
28446
7800
different methods are compared, as shown in Fig. 4. The results by the IE-CSD-ODDM are well consistent with that by the IE-ODDM. Table I shows that the IE-CSD-ODDM can reduce about 30.94% from the memory requirement in the IEODDM, and Table II shows that the IE-CSD-ODDM can reduce about 26.44% from the filling time in the IE-ODDM.
Fig. 4. The RCS of a PEC sphere, obtained by the IE-ODDM and the IECSD-ODDM respectively.
Example III: A PEC missile-type object is considered, which consists of 13 sub-domains. All the 13 sub-domains are classified into 2 types of congruent sub-domains. Detailedly, the first type congruent sub-domain has 10 sub-domains, the second type congruent sub-domain has 2 sub-domains, and the un-congruent sub-domain is only one sub-domain, as shown in Fig. 5. The object has the height 11O and the bottom radius O , and involves 28446 unknowns. The results by the two different methods are compared, as shown in Fig. 5. The results by the IE-CSD-ODDM are well consistent with that by the IE-ODDM. Table I shows that the IE-CSD-ODDM can reduce about 50.86% from the memory requirements in the IEODDM, and Table II shows that the IE-CSD-ODDM can reduce about 29.67% from the filling time in the IE-ODDM.
consists of 22 sub-domains. All the 22 sub-domains are classified into 5 types of congruent sub-domains. Detailedly, the first type congruent sub-domain has 8 sub-domains, all the other type congruent sub-domains have 2 sub-domains, and the un-congruent sub-domains are 5 sub-domains, as shown in Fig. 5. The object has the body length 6.4O , the cabin diameter 0.6O and the wing length 2.6O , and involves 7800 unknowns. The results by the two different methods are compared, as shown in Fig. 6. The results by the IE-CSDODDM are well consistent with that by the IE-ODDM. Table I shows that the IE-CSD-ODDM can reduce about 17.20% from the memory requirements in the IE-ODDM, and Table II shows that the IE-CSD-ODDM can reduce about 12.08% from the filling time in the IE-ODDM.
Example IV: A PEC airplane-type object is considered, which
More details can be found in Table III and Table IV. Table
Fig. 5. The RCS of a PEC missile-type object, obtained by the IE-ODDM and the IE-CSD-ODDM respectively.
Fig. 3. The RCS of a PEC cube, obtained by the IE-ODDM and the IE-CSDODDM respectively.
Specific Research Program of China (No. 201110046-2), and in part by the National High-Tech Research Plan (863) of China (No. 2008AA01Z223). REFERENCES [1] [2]
[3]
[4]
[5]
[6] Fig. 6. The RCS of a PEC airplane-type object, obtained by the IE-ODDM and the IE-CSD-ODDM respectively. [7]
III lists the solving time of each scanning for all the subdomains, and Table IV lists the primary parameters of the four examples. From Table III, one can find that the iteration times of each round of scanning are roughly the same for the IECSD-ODDM and the IE-ODDM.
[8]
[9]
IV. CONCLUSION [10]
This paper develops an overlapped domain decomposition method with congruent sub-domains for solving EM surface integral equation. In this method, a uniform meshing scheme is employed such that the self-matrix on each sub-domain share a common sub-matrix, which results in significant reduction of the storage and the matrix filling time, especially for those surface structures with a lot of large congruent subdomains. The future work is to build a hybrid method based the MLFMA and the IE-CSD-ODDM, which will not only accelerate the solving time but also reduce the memory requirement and the filling time of matrices intensively.
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ACKNOWLEDGMENT This work was supported in part by the National Basic Research Program of China (No. 2009CB320203, and No. 2010CB327400), in part by the Science Fund for Creative Research Groups of China (No. 60921063), in part by the National Science Fund for Young Scholars of China (No. 60901013), and in part by the National Non-profit Industry
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