# Acceleration of Augmented EFIE Using Multilevel Complex Source Beam Method

Authors: Lianning Song; Yongpin Chen; Ming Jiang; Jun Hu; Zaiping Nie
Journal: International Journal of Antennas and Propagation
 (2017)

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## Abstract

The computation of the augmented electric field integral equation (A-EFIE) is accelerated by using the multilevel complex source beam (MLCSB) method. As an effective solution of the low-frequency problem, A-EFIE includes both current and charge as unknowns to avoid the imbalance between the vector potentials and the scalar potentials in the conventional EFIE. However, dense impedance submatrices are involved in the A-EFIE system, and the computational cost becomes extremely high for problems with a large number of unknowns. As an exact solution to Maxwell’s equations, the complex source beam (CSB) method can be well tailored for A-EFIE to accelerate the matrix-vector products in an iterative solver. Different from the commonly used multilevel fast multipole algorithm (MLFMA), the CSB method is free from the problem of low-frequency breakdown. In our implementation, the expansion operators of CSB are first derived for the vector potentials and the scalar potentials. Consequently, the aggregation and disaggregation operators are introduced to form a multilevel algorithm to reduce the computational complexity. The accuracy and efficiency of the proposed method are discussed in detail through a variety of numerical examples. It is observed that the numerical error of the MLCSB-AEFIE keeps constant for a broad frequency range, indicating the good stability and scalability of the proposed method.


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## Body

## 1. Introduction

The method of moments (MoM) [1] for solving electric field integral equation (EFIE) has received intensive study in the analysis of electromagnetic (EM) radiation, scattering, and circuit problems in recent years. However, the commonly used EFIE with RWG basis function suffers from a “low-frequency breakdown” problem [2] when the frequency decreases and/or when the mesh is refined. In this situation, EFIE is dominated by the scalar potentials while the contribution from the vector potentials is overwhelmed due to the finite machine precision. Since the scalar potential term is singular, the EFIE matrix system becomes extremely ill-conditioned and results in convergence issue when the iterative solver is applied.To address this difficulty, several methods have been proposed in the past years. A widely used remedy is the loop-tree or loop-star decomposition [3] which decouples the magneto- and electrostatic physics. However, the extraction of global loops is difficult for complex interconnecting structures, and the system matrix becomes ill-conditioned as the frequency increases. On the other hand, the convergence issue can be improved by a Calderón preconditioner [4] when iterative solvers are applied. To avoid the involved loop searching process, alternative formulations have been proposed by separating current and charge, such as the current and charge integral equation (CCIE) [5], split potential integral equation (SPIE) [6, 7], and augmented electric field integral equation (A-EFIE) [8–11]. In the A-EFIE method, the contributions of the vector potentials and the scalar potentials are separated by adding charge as extra unknowns, where the current continuity equation is explicitly enforced. Frequency scaling is then implemented to stabilize the system equation. The resulting impendence matrix of A-EFIE has the characteristics of a saddle point matrix and is well-conditioned at low frequencies. Recently, the perturbation method is introduced to enhance the accuracy of A-EFIE at extremely low frequencies [12]. The augmented equivalence principle algorithm (A-EPA) [13] and discontinuous Galerkin (DG) method [14] are combined with the A-EFIE for domain decomposition problems.As the number of unknowns increases, a fast algorithm has to be incorporated into the iterative solver to reduce the operation complexity of the matrix-vector product (MVP). Such fast algorithms should be low-frequency stable; for instance, the low-frequency multilevel fast multipole algorithm (LF-MLFMA) [15], the multilevel accelerated Cartesian expansion algorithm (MLACEA) [16], the multilevel adaptive cross-approximation (MLACA) algorithm [17], and fast Fourier transform (FFT) [18] are eligible candidates for such purpose. Recently, a complex source beam-method of moment (CSB-MoM) is proposed to accelerate the far-field interactions of MoM at midfrequencies [19]. The object is first divided into groups and complex source beams (CSBs) are used to expand the fields of the basis functions residing in each group [20, 21]. A multilevel version of this method is developed in [22]. To further improve the computational efficiency, a nested complex source beam (NCSB) method is proposed by utilizing an equivalent relationship between adjacent levels [23]. Since the CSBs are exact solutions of Maxwell’s equations, any arbitrary EM fields can be expanded in terms of a set of CSBs [20, 24]. Therefore, this method can be extended to solve the low-frequency problems without any theoretical barriers.This paper is organized as follows. The basis formulations of CSB-MoM and A-EFIE are briefly reviewed in Section2. In Section 3, the CSB-MoM is integrated into the A-EFIE system to remedy the low-frequency breakdown problem. The detailed derivation of the CSB expansions for A-EFIE is first presented. The aggregation/disaggregation, translation operators for a multilevel CSB method are then discussed. Finally, numerical examples are summarized in Section 4 to demonstrate the validity and efficiency of the proposed method.


## 2. Theory Background

Given a 3D perfectly electrical conducting (PEC) body defined by its surface, the conventional MoM formulation can be applied to the electric field integral equation (EFIE), leading to a matrix equation of the form(1)Zj=b,where j is the unknown vector for the surface current density, b is the excitation vector, and Z is the dense impedance matrix. The impedance matrix can be expressed as(2)Zij=∫Sifir·∫SjG̿r,r′fjr′dS′,where(3)G̿r,r′=I̿+∇∇k02gr,r′is the dyadic Green function andg(r,r′)=eik0|r-r′|/4π|r-r′| is the scalar Green function. Moreover,fj(r′)andfi(r)denote the RWG basis functions for expanding surface current and Galerkin testing, respectively. This equation suffers from the low-frequency breakdown problem, because the contribution of the vector potential is swamped by that of the scalar potential at low frequencies due to the finite machine precision.


### 2.1. Formulation of CSB-MoM for EFIE

In CSB-MoM method, the MVP acceleration ofZj=b is achieved through a FMM-like near-field and far-field decomposition. For the far-field interactions ZFj between well separated groups, the MVP are carried out by a series of CSBs launched on a complex equivalence surface enclosing each group:(4)ZFj=Zmm′jm′=∑ε,ε′=θ,ϕWmεTTL,mm′εε′Wm′ε′jm′,where m and m′ denote the observation and source groups and L indicates the finest level (single level in CSB-MoM). Wm′ε and Wmε′ are the expansion matrices for both θ and ϕ components, which can be obtained by the far-field matching of basis functions in groups m and m′, respectively [19]. Superscript T stands for the transpose of the corresponding matrix. It needs to be emphasized that only θ and ϕ components are considered in CSBs, since the far-field radiation fields contain only θ and ϕ components.By using the expansion matrices, the CSB expansion coefficients for the source groupm′ can be expanded from the surface currents as(5)sL,m′=sL,m′θsL,m′ϕ=Wm′θWm′ϕjm′.T
L
,
m
m
′
ε
ε
′ in (4) is the translation matrix, of which the elements are expressed as(6)TL,mm′εε′qq′=ε^q·G̿r~L,mq,r~L,m′′q′·ε^q′′,where ε^q and ε^q′′ denote the unit vectors of qth and q′th CSB, respectively. The complex position vectors r~L,mq and r~L,m′′q′ are the launch points of CSBs associated with groups m and m′. G̿(r~L,mq,r~L,m′′q′) is the dyadic Green function with complex arguments. The directional property of CSBs can be used to reduce the computational cost in the translation procedure.


### 2.2. Formulation of A-EFIE

In the augmented electric field integral equation, the surface current is discretized by the normalized RWG basis function, which is modified by removing the length of the common edge(7)ρir=r-ri+2Ai+,r∈Ti+ri--r2Ai-,r∈Ti-0,otherwise,where Ai± are the area of trianglesTi±, ri±are the free vertices of the two triangles. On the other hand, the surface charge density is approximated by the pulse function defined on each triangle, which is expressed as(8)hir=1Ai,r∈Ti0,otherwise.Combining EFIE in (1) and the current continuity condition between current and charge, we can arrive at the A-EFIE equation(9)V¯D¯T·P¯D¯k02I¯·ik0jc0ϕ=η0-1b0with(10)V¯ij=∫Siρir·∫Sjgr,r′ρjr′dS′dS(11)P¯ij=∫Sihir·∫Sjgr,r′hjr′dS′dS(12)bi=∫Sifir·EidS(13)D¯ij=1,Ti∈Sj,thepositivepart-1,Ti∈Sj,thenegativepart0,Ti∉Sj,where j and ϕ denote the unknown coefficients for current and charge, k0 and η0 are the wave number and the wave impedance in free space, andc0is the light speed in free space. The dense matrix V¯ represents the vector potentials and depicts the current interactions between inner edges. On the other hand, P¯ denotes the scalar potentials, which describes charges interactions between triangles. The sparse matrix D¯ represents the relationship between edge and the triangles patch, and I¯ is the identity matrix. In the matrix system (9), the vector potential and scalar potential are balanced by using a proper frequency scaling, which is critical for low-frequency problems.


## 2.1. Formulation of CSB-MoM for EFIE

In CSB-MoM method, the MVP acceleration ofZj=b is achieved through a FMM-like near-field and far-field decomposition. For the far-field interactions ZFj between well separated groups, the MVP are carried out by a series of CSBs launched on a complex equivalence surface enclosing each group:(4)ZFj=Zmm′jm′=∑ε,ε′=θ,ϕWmεTTL,mm′εε′Wm′ε′jm′,where m and m′ denote the observation and source groups and L indicates the finest level (single level in CSB-MoM). Wm′ε and Wmε′ are the expansion matrices for both θ and ϕ components, which can be obtained by the far-field matching of basis functions in groups m and m′, respectively [19]. Superscript T stands for the transpose of the corresponding matrix. It needs to be emphasized that only θ and ϕ components are considered in CSBs, since the far-field radiation fields contain only θ and ϕ components.By using the expansion matrices, the CSB expansion coefficients for the source groupm′ can be expanded from the surface currents as(5)sL,m′=sL,m′θsL,m′ϕ=Wm′θWm′ϕjm′.T
L
,
m
m
′
ε
ε
′ in (4) is the translation matrix, of which the elements are expressed as(6)TL,mm′εε′qq′=ε^q·G̿r~L,mq,r~L,m′′q′·ε^q′′,where ε^q and ε^q′′ denote the unit vectors of qth and q′th CSB, respectively. The complex position vectors r~L,mq and r~L,m′′q′ are the launch points of CSBs associated with groups m and m′. G̿(r~L,mq,r~L,m′′q′) is the dyadic Green function with complex arguments. The directional property of CSBs can be used to reduce the computational cost in the translation procedure.


## 2.2. Formulation of A-EFIE

In the augmented electric field integral equation, the surface current is discretized by the normalized RWG basis function, which is modified by removing the length of the common edge(7)ρir=r-ri+2Ai+,r∈Ti+ri--r2Ai-,r∈Ti-0,otherwise,where Ai± are the area of trianglesTi±, ri±are the free vertices of the two triangles. On the other hand, the surface charge density is approximated by the pulse function defined on each triangle, which is expressed as(8)hir=1Ai,r∈Ti0,otherwise.Combining EFIE in (1) and the current continuity condition between current and charge, we can arrive at the A-EFIE equation(9)V¯D¯T·P¯D¯k02I¯·ik0jc0ϕ=η0-1b0with(10)V¯ij=∫Siρir·∫Sjgr,r′ρjr′dS′dS(11)P¯ij=∫Sihir·∫Sjgr,r′hjr′dS′dS(12)bi=∫Sifir·EidS(13)D¯ij=1,Ti∈Sj,thepositivepart-1,Ti∈Sj,thenegativepart0,Ti∉Sj,where j and ϕ denote the unknown coefficients for current and charge, k0 and η0 are the wave number and the wave impedance in free space, andc0is the light speed in free space. The dense matrix V¯ represents the vector potentials and depicts the current interactions between inner edges. On the other hand, P¯ denotes the scalar potentials, which describes charges interactions between triangles. The sparse matrix D¯ represents the relationship between edge and the triangles patch, and I¯ is the identity matrix. In the matrix system (9), the vector potential and scalar potential are balanced by using a proper frequency scaling, which is critical for low-frequency problems.


## 3. Implementation of MLCSB for A-EFIE

### 3.1. CSB Expansions for A-EFIE

Motivated by the idea of CSB expansion method for EFIE, we can derive the expansions for A-EFIE in a similar way. The inner integrals of (10) and (11) can be written as (14)Vir=∫Sigr,r′ρir′dS′(15)Pjr=∫Sjgr,r′hjr′dS′,where i and j represent the ith normalized RWG basis function and the jth pulse function in group m′. The above integral equation is equivalent to the summation of CSB vector and scalar potentials: (16)Vir=∑q=1Qgr,r~q′x^wq,ix+y^wq,iy+z^wq,iz(17)Pjr=∑q=1Qgr,r~q′wq,jp,where x^wq,ix+y^wq,iy+z^wq,iz is the vector CSB weight for current i, wq,jp is the weight for chargej, and Q represents the total number of beams in this discretization. Combining (16) with (14) and (17) with (15) and testing on a far-field matching point, we get(18)x^·∑q=1Qgrt,r~q′x^wq,ix=x^·∫Sigrt,r′ρir′dS′y^·∑q=1Qgrt,r~q′y^wq,iy=y^·∫Sigrt,r′ρir′dS′z^·∑q=1Qgrt,r~q′z^wq,iz=z^·∫Sigrt,r′ρir′dS′∑q=1Qgrt,r~q′wq,jp=∫Sjgrt,r′hjr′dS′,where rt (t=1,2,…,Q) is the matching point. Those linear equations can be converted into a matrix form and combined as a multiple right-hand side problem by all the current and charge basis in the group m′:(19)GWm′x,Wm′y,Wm′z,Wm′p=Vm′x,Vm′y,Vm′z,Pm′,where(20)Gtq=grt,r~q′Wm′vqi=wq,ivv=x,y,zWm′pqj=wq,jpVm′vti=v^t·∫Sigrt,r′ρir′dS′v=x,y,zPm′tj=∫Sjgrt,r′hjr′dS′.By using the expansion matricesWm′x,Wm′y,Wm′z, and Wm′p, the CSB expansion coefficients for vector potentials of group m′ can be expanded from jm′ as(21)s¯L,m′v=sL,m′x,sL,m′y,sL,m′z=Wm′x,Wm′y,Wm′zjm′.Similarly, the coefficients for scalar ones are(22)sL,m′p=Wm′pϕm′.Hence, the far-field MVP procedure for (10) and (11) can be obtained with a similar process in (4):(23)V¯mm′jm′=∑v=x,y,zWmvTTL,mm′Wm′vjm′P¯mm′ϕm′=WmdTTL,mm′Wm′pϕm′,where the translation operator TL,mm′ is expressed as(24)TL,mm′qq′=gr~L,mq,r~L,m′′q′.It is noticed that the scalar Green function with complex arguments is used here, and the translation for the vector and scalar expansion coefficients shares the same operators.


### 3.2. Multilevel Algorithm of CSB for A-EFIE

So far, we have derived the CSB expansion method for A-EFIE and have presented the far-group MVP process with the help of translation operators in a single level algorithm. In the following, we will obtain the aggregation and disaggregation operators of CSB expansion coefficients for both (10) and (11), to realize a multilevel algorithm. Firstly, the vector and scalar potentials by the ql+1th CSB in a group is (25)Vql+1r=gr,r~ql+1′x^sql+1x+y^sql+1y+z^sql+1zPql+1r=gr,r~ql+1′sql+1pwhere l+1 means the (l+1)th level of the octree in the multilevel algorithm. By applying the summation and testing procedures similar to (16)–(18), a linear system can be set up to obtain the equivalent relationship of CSBs between two adjacent levels:(26)GlAl=Pl+1,where Gltlql=g(rtl,r~ql′) is the matching matrix in parent level l. This matching matrix connects the equivalent CSB sources to the scalar potentials Pl+1. Different from the right-hand side in (19), the right-hand side here for each vector potential component is the same as the scalar one, which is(27)Pl+1qlql+1=grql∣r~ql+1′.OnceGl and Pl+1 are assembled, the aggregation matrix Al for level l can be numerically solved. By using the aggregation matrix, the CSB expansion coefficients of the parent group in levellcan be obtained efficiently from its child groups in levell + 1: (28)s¯l,m′v=sl,m′x,sl,m′y,sl,m′z=∑n′∈childm′Als¯l+1,n′vsl,m′p=∑n′∈childm′Alsl+1,n′p.Similar to the multilevel fast multipole algorithm (MLFMA) [25], the CSB expansion coefficients of a receiving group in level l+1 are obtained from the translation in the same level as well as the disaggregation from its parent level l. The disaggregation matrix can be easily obtained from the transpose of the aggregation matrices.


## 3.1. CSB Expansions for A-EFIE

Motivated by the idea of CSB expansion method for EFIE, we can derive the expansions for A-EFIE in a similar way. The inner integrals of (10) and (11) can be written as (14)Vir=∫Sigr,r′ρir′dS′(15)Pjr=∫Sjgr,r′hjr′dS′,where i and j represent the ith normalized RWG basis function and the jth pulse function in group m′. The above integral equation is equivalent to the summation of CSB vector and scalar potentials: (16)Vir=∑q=1Qgr,r~q′x^wq,ix+y^wq,iy+z^wq,iz(17)Pjr=∑q=1Qgr,r~q′wq,jp,where x^wq,ix+y^wq,iy+z^wq,iz is the vector CSB weight for current i, wq,jp is the weight for chargej, and Q represents the total number of beams in this discretization. Combining (16) with (14) and (17) with (15) and testing on a far-field matching point, we get(18)x^·∑q=1Qgrt,r~q′x^wq,ix=x^·∫Sigrt,r′ρir′dS′y^·∑q=1Qgrt,r~q′y^wq,iy=y^·∫Sigrt,r′ρir′dS′z^·∑q=1Qgrt,r~q′z^wq,iz=z^·∫Sigrt,r′ρir′dS′∑q=1Qgrt,r~q′wq,jp=∫Sjgrt,r′hjr′dS′,where rt (t=1,2,…,Q) is the matching point. Those linear equations can be converted into a matrix form and combined as a multiple right-hand side problem by all the current and charge basis in the group m′:(19)GWm′x,Wm′y,Wm′z,Wm′p=Vm′x,Vm′y,Vm′z,Pm′,where(20)Gtq=grt,r~q′Wm′vqi=wq,ivv=x,y,zWm′pqj=wq,jpVm′vti=v^t·∫Sigrt,r′ρir′dS′v=x,y,zPm′tj=∫Sjgrt,r′hjr′dS′.By using the expansion matricesWm′x,Wm′y,Wm′z, and Wm′p, the CSB expansion coefficients for vector potentials of group m′ can be expanded from jm′ as(21)s¯L,m′v=sL,m′x,sL,m′y,sL,m′z=Wm′x,Wm′y,Wm′zjm′.Similarly, the coefficients for scalar ones are(22)sL,m′p=Wm′pϕm′.Hence, the far-field MVP procedure for (10) and (11) can be obtained with a similar process in (4):(23)V¯mm′jm′=∑v=x,y,zWmvTTL,mm′Wm′vjm′P¯mm′ϕm′=WmdTTL,mm′Wm′pϕm′,where the translation operator TL,mm′ is expressed as(24)TL,mm′qq′=gr~L,mq,r~L,m′′q′.It is noticed that the scalar Green function with complex arguments is used here, and the translation for the vector and scalar expansion coefficients shares the same operators.


## 3.2. Multilevel Algorithm of CSB for A-EFIE

So far, we have derived the CSB expansion method for A-EFIE and have presented the far-group MVP process with the help of translation operators in a single level algorithm. In the following, we will obtain the aggregation and disaggregation operators of CSB expansion coefficients for both (10) and (11), to realize a multilevel algorithm. Firstly, the vector and scalar potentials by the ql+1th CSB in a group is (25)Vql+1r=gr,r~ql+1′x^sql+1x+y^sql+1y+z^sql+1zPql+1r=gr,r~ql+1′sql+1pwhere l+1 means the (l+1)th level of the octree in the multilevel algorithm. By applying the summation and testing procedures similar to (16)–(18), a linear system can be set up to obtain the equivalent relationship of CSBs between two adjacent levels:(26)GlAl=Pl+1,where Gltlql=g(rtl,r~ql′) is the matching matrix in parent level l. This matching matrix connects the equivalent CSB sources to the scalar potentials Pl+1. Different from the right-hand side in (19), the right-hand side here for each vector potential component is the same as the scalar one, which is(27)Pl+1qlql+1=grql∣r~ql+1′.OnceGl and Pl+1 are assembled, the aggregation matrix Al for level l can be numerically solved. By using the aggregation matrix, the CSB expansion coefficients of the parent group in levellcan be obtained efficiently from its child groups in levell + 1: (28)s¯l,m′v=sl,m′x,sl,m′y,sl,m′z=∑n′∈childm′Als¯l+1,n′vsl,m′p=∑n′∈childm′Alsl+1,n′p.Similar to the multilevel fast multipole algorithm (MLFMA) [25], the CSB expansion coefficients of a receiving group in level l+1 are obtained from the translation in the same level as well as the disaggregation from its parent level l. The disaggregation matrix can be easily obtained from the transpose of the aggregation matrices.


## 4. Numerical Results

In this section, the accuracy, error analysis, computational complexity, and the efficiency of the method are investigated through several numerical examples. All the examples were run on a computer of 2 processors, each with 14 cores at 2.6 GHz, 512 GB memory, and OpenMP parallelization.


### 4.1. Small Sphere

To show the accuracy of MLCSB-AEFIE at low frequencies, the electromagnetic scattering by a PEC sphere of 1 m radius is analyzed at 100 Hz. The sphere is discretized with 1,764 triangular patches, which corresponds to 2,646 inner edges. A three levels’ MLCSB algorithm is used with a group size of8.3×10-8λ at the finest level. The incident angle of a plane wave is θi=0°,φi=0°, and the observed azimuth angle is fixed at θi=0°. The residual error threshold is set to be 10-15 for GMRES-30. It takes 80 iteration steps to converge, with the help of the saddle point preconditioner in [9]. A good agreement of the bistatic RCS is observed in Figure 1, as compared with the analytical solution of Mie series.Figure 1
Bistatic RCS of the PEC sphere at 100 Hz, validated by the Mie series.


### 4.2. Computational Complexity

To demonstrate the computational complexity of the proposed method, the plane wave scattering of a PEC cube with a side length of 0.1 m is calculated at 300 MHz. The electric size of the cube is0.173λ. The surface of the cube is discretized into six different meshes: Mesh A, Mesh B, Mesh C, Mesh D, Mesh E, and Mesh F. The coarsest Mesh A has 1262 planar triangles and 1893 interior edges, and the average edge length is 1.05×10-2λ. Then we refine Mesh A by halving the edge length recursively until Mesh F. Mesh F comprises 1,534,536 planar triangles and 2,301,804 interior edges, and the average edge length is 3×10-4λ.The CPU times for each MVP and the memory usage is plotted in Figure2. It can be found that the computational complexity and the memory requirement of MLCSB-AEFIE both scale as O(N), where N are the unknown numbers. The detailed computational statistics are summarized in Table 1, and the iteration histories are shown in Figure 3 for comparison. By using the preconditioner in [9], the iteration converges quickly to 10-4. A good agreement of the bistatic radar cross sections (RCS) of the five cases is demonstrated in Figure 4.Table 1
Computational statistics of the PEC cube with different meshes at 300 MHz.
Mesh
Number of RWG
Number of triangles
Near-field Mem. (MB)
Expansion Mem. (MB)
Translation Mem. (MB)
Interpolation Mem. (MB)
Total Mem. (MB)
MVP time (s)
Iteration number
Total time (s)
A
1,893
1,262
2.7
21.5
59.1
4.2
87.5
0.15
25
4.1
B
7,992
5,328
10.2
90.7
88.6
8.4
197.9
0.6
36
22.8
C
33,330
22,220
43.5
378.3
118.2
12.6
552.6
2.2
52
118.4
D
139,002
92,668
178
1,541
147.8
16.8
1,883.6
8.7
74
655.8
E
563,292
375,528
727.9
6,245
177.36
21
7,170.4
31.2
110
3,480
F
2,301,804
1,534,536
2,932.9
25,518
206.9
25.2
28,683
120.9
174
21,232.6Figure 2
CPU time for each MVP and memory usage in the example of the PEC cube.Figure 3
Iteration history of the PEC cube for different meshes.Figure 4
Bistatic RCS of the PEC cube with different meshes.


### 4.3. Error Analysis

In the proposed MLCSB-AEFIE, the numerical error is mainly from expansion and aggregation. Since the expansion and aggregation processes are similar for the vector and scalar potentials, we only show the scalar potential case in the following. For the expansion, the relative errors related to a group with the size of 0.005 m and average discretization length of 0.001 m are studied. The relative error of expansion is defined as(29)e=Pr-PCSBrPr,where PCSB(r) is the scalar potential obtained via the expansion process in (17), P(r) is the exact data calculated by (15), and the observation point r is fixed at the center of the nearest cousin group. Figure 5 shows the relative errorsefor different numbers of CSBs Q=30, 42, 58, and 82 with respect to the frequencies from 30 Hz to 300 MHz. It is noticed that the expansion errors are almost constant for the same Q in a very wide frequency range, indicating an excellent stability and scalability of the proposed method for broadband computations.Figure 5
Relative errors of expansion.To investigate the interpolation error in the aggregation process, we compare the scalar potentials related to a group with the size of 0.005 m and the parent group with the size of 0.01 m. The scalar potentials are calculated by CSBs, while the CSB coefficients of the parent group are aggregated from the child group. The RMS relative error is shown in Figure6, which is defined as(30)RMSP=1N∑i=1NPlCSBri-Pl+1CSBri2Pl+1CSBri2and here PlCSB(ri),  Pl+1CSB(ri) are the scalar potentials of the parent group and the child group, respectively, and ri is set on azimuth circle centered at the parent group. Figure 6 also shows a constant accuracy level for different frequencies with the same Q.Figure 6
RMS errors of interpolation.


### 4.4. Scattering of Multiscale Cone

The performance of the MLCSB-AEFIE for multiscale structures is evaluated by a nonuniformly meshed cone illustrated in Figure7. The radius of the cone is 5 m at the bottom and the height is 10 m. We mesh the cone with a length of 0.1 m at the bottom and gradually reduce it to 0.001 m at the sharp point. Finally, the cone is discretized with 191,976 planar patches, and hence 287,964 interior edges. Figure 8 shows the bistatic RCS of the cone excited by a y-polarized plane wave incident from thexdirection at10KHz and10MHz. In this example, MLCSB-AEFIE converges to relative residual error of 10-3 within 73 and 67 iterations, for 10 KHz and 10 MHz,respectively.Figure 7
A cone of radius 5 m and height 10 m, which is meshed with average length 0.1 m at the bottom and 0.001 m at the sharp point. Mesh details are shown with reference lengths 0.1 m and 0.01 m.Figure 8
Bistatic RCS of the PEC cone at 10 KHz and 10 Mhz.


## 4.1. Small Sphere

To show the accuracy of MLCSB-AEFIE at low frequencies, the electromagnetic scattering by a PEC sphere of 1 m radius is analyzed at 100 Hz. The sphere is discretized with 1,764 triangular patches, which corresponds to 2,646 inner edges. A three levels’ MLCSB algorithm is used with a group size of8.3×10-8λ at the finest level. The incident angle of a plane wave is θi=0°,φi=0°, and the observed azimuth angle is fixed at θi=0°. The residual error threshold is set to be 10-15 for GMRES-30. It takes 80 iteration steps to converge, with the help of the saddle point preconditioner in [9]. A good agreement of the bistatic RCS is observed in Figure 1, as compared with the analytical solution of Mie series.Figure 1
Bistatic RCS of the PEC sphere at 100 Hz, validated by the Mie series.


## 4.2. Computational Complexity

To demonstrate the computational complexity of the proposed method, the plane wave scattering of a PEC cube with a side length of 0.1 m is calculated at 300 MHz. The electric size of the cube is0.173λ. The surface of the cube is discretized into six different meshes: Mesh A, Mesh B, Mesh C, Mesh D, Mesh E, and Mesh F. The coarsest Mesh A has 1262 planar triangles and 1893 interior edges, and the average edge length is 1.05×10-2λ. Then we refine Mesh A by halving the edge length recursively until Mesh F. Mesh F comprises 1,534,536 planar triangles and 2,301,804 interior edges, and the average edge length is 3×10-4λ.The CPU times for each MVP and the memory usage is plotted in Figure2. It can be found that the computational complexity and the memory requirement of MLCSB-AEFIE both scale as O(N), where N are the unknown numbers. The detailed computational statistics are summarized in Table 1, and the iteration histories are shown in Figure 3 for comparison. By using the preconditioner in [9], the iteration converges quickly to 10-4. A good agreement of the bistatic radar cross sections (RCS) of the five cases is demonstrated in Figure 4.Table 1
Computational statistics of the PEC cube with different meshes at 300 MHz.
Mesh
Number of RWG
Number of triangles
Near-field Mem. (MB)
Expansion Mem. (MB)
Translation Mem. (MB)
Interpolation Mem. (MB)
Total Mem. (MB)
MVP time (s)
Iteration number
Total time (s)
A
1,893
1,262
2.7
21.5
59.1
4.2
87.5
0.15
25
4.1
B
7,992
5,328
10.2
90.7
88.6
8.4
197.9
0.6
36
22.8
C
33,330
22,220
43.5
378.3
118.2
12.6
552.6
2.2
52
118.4
D
139,002
92,668
178
1,541
147.8
16.8
1,883.6
8.7
74
655.8
E
563,292
375,528
727.9
6,245
177.36
21
7,170.4
31.2
110
3,480
F
2,301,804
1,534,536
2,932.9
25,518
206.9
25.2
28,683
120.9
174
21,232.6Figure 2
CPU time for each MVP and memory usage in the example of the PEC cube.Figure 3
Iteration history of the PEC cube for different meshes.Figure 4
Bistatic RCS of the PEC cube with different meshes.


## 4.3. Error Analysis

In the proposed MLCSB-AEFIE, the numerical error is mainly from expansion and aggregation. Since the expansion and aggregation processes are similar for the vector and scalar potentials, we only show the scalar potential case in the following. For the expansion, the relative errors related to a group with the size of 0.005 m and average discretization length of 0.001 m are studied. The relative error of expansion is defined as(29)e=Pr-PCSBrPr,where PCSB(r) is the scalar potential obtained via the expansion process in (17), P(r) is the exact data calculated by (15), and the observation point r is fixed at the center of the nearest cousin group. Figure 5 shows the relative errorsefor different numbers of CSBs Q=30, 42, 58, and 82 with respect to the frequencies from 30 Hz to 300 MHz. It is noticed that the expansion errors are almost constant for the same Q in a very wide frequency range, indicating an excellent stability and scalability of the proposed method for broadband computations.Figure 5
Relative errors of expansion.To investigate the interpolation error in the aggregation process, we compare the scalar potentials related to a group with the size of 0.005 m and the parent group with the size of 0.01 m. The scalar potentials are calculated by CSBs, while the CSB coefficients of the parent group are aggregated from the child group. The RMS relative error is shown in Figure6, which is defined as(30)RMSP=1N∑i=1NPlCSBri-Pl+1CSBri2Pl+1CSBri2and here PlCSB(ri),  Pl+1CSB(ri) are the scalar potentials of the parent group and the child group, respectively, and ri is set on azimuth circle centered at the parent group. Figure 6 also shows a constant accuracy level for different frequencies with the same Q.Figure 6
RMS errors of interpolation.


## 4.4. Scattering of Multiscale Cone

The performance of the MLCSB-AEFIE for multiscale structures is evaluated by a nonuniformly meshed cone illustrated in Figure7. The radius of the cone is 5 m at the bottom and the height is 10 m. We mesh the cone with a length of 0.1 m at the bottom and gradually reduce it to 0.001 m at the sharp point. Finally, the cone is discretized with 191,976 planar patches, and hence 287,964 interior edges. Figure 8 shows the bistatic RCS of the cone excited by a y-polarized plane wave incident from thexdirection at10KHz and10MHz. In this example, MLCSB-AEFIE converges to relative residual error of 10-3 within 73 and 67 iterations, for 10 KHz and 10 MHz,respectively.Figure 7
A cone of radius 5 m and height 10 m, which is meshed with average length 0.1 m at the bottom and 0.001 m at the sharp point. Mesh details are shown with reference lengths 0.1 m and 0.01 m.Figure 8
Bistatic RCS of the PEC cone at 10 KHz and 10 Mhz.


## 5. Conclusion

In this paper, we have proposed a MLCSB-AEFIE method for the well-known low-frequency problem. The vector and scalar potentials from the current and charge unknowns are expended with CSBs. An aggregation matrix is obtained for the CSB expansion coefficients to form an efficient multilevel algorithm. Numerical examples have validated the good accuracy, efficiency, and scalability of the proposed method for low-frequency problems.



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*Source: 9640136-2017-07-13.xml*