基于域分解的混合求解器在层状介质中的复合散射资源-CSDN文库

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本篇研究论文的标题是“基于域分解的混合求解器在层状介质中的复合散射”。在此标题下，涉及的关键技术点和概念主要有以下几个方面： 1. 域分解方法（Domain Decomposition Method, DDM）：域分解方法是一种在电磁工程领域内广受欢迎的研究主题。它通常将一个复杂的计算域分割成若干个更小的子域，并分别对这些子域进行求解。这种方法允许对计算过程进行并行处理，从而提升计算效率。论文提到，DDM在自由空间的应用已经得到了广泛的研究，但在层状介质（Layered Medium, LM）中的应用却鲜有报道。 2. 层状介质（Layered Medium, LM）：层状介质指的是由不同材料层组成的介质，每层可能具有不同的电磁特性。在层状介质中进行电磁仿真时，考虑不同层之间的相互作用和波的传播特性是非常重要的。由于环境的复杂性，这通常会增加仿真计算的难度。 3. 表面积分方程（Surface Integral Equation, SIE）与有限元方法（Finite Element Method, FEM）的结合：表面积分方程方法通常被用来模拟表面域，而有限元方法则常用于求解体积域。在本篇论文中，提出了基于DDM的混合SIE/FEM求解器，可以有效地最小化仿真区域。通过这种方法，考虑到了不均匀背景的影响，并通过Riemann求解器在不同子域界面之间实现场的传输。 4. Riemann求解器边界条件（Riemann Solver Boundary Condition）：这一边界条件用于连接不同子域界面处的场，使得各个子域之间能够正确地传递电磁信息，是实现域分解方法中子域之间耦合的关键技术之一。 5. Krylov子空间求解器（Krylov Subspace Solver）与预处理方法（Preconditioning Method）：这一算法用于求解由此产生的矩阵方程。Krylov子空间求解器在处理大型稀疏矩阵问题时表现优异，尤其适用于电磁场仿真计算。预处理方法则可以改善矩阵的条件数，使得求解过程更加稳定和快速。 6. 数值结果的验证（Numerical Results Validation）：为了验证所提出的混合求解器的准确性，文中给出了相应的数值结果。这一步是检验新算法是否能够有效地解决层状介质中的复合散射问题的重要步骤。综合上述内容，这篇研究论文探讨了利用混合求解器结合表面积分方程方法和有限元方法来求解层状介质中复合物体的电磁仿真问题。这需要将计算域分割成表面域和多个体积子域，表面域通过表面积分方程和层状介质的格林函数（LMGFs）进行建模，而体积子域则通过有限元方法进行建模。在子域之间，使用了Riemann求解器作为边界条件来连接各子域界面的场。最终，通过Krylov子空间求解器和预处理方法来求解得到的矩阵方程，数值结果则用于验证所提求解器的准确性。论文指出，DDM方法虽然在自由空间应用广泛，但在层状介质中的应用则相对较为罕见，因此本研究提出的基于非共形网格的混合FEM/SIE求解器方案更具有吸引力。

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IEEE Proof

IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 00, NO. 00, 2016 1

A Hybrid Solver Based on Domain Decomposition

Method for the Composite Scattering

in Layered Medium

Yi Ren, Member, IEEE, Wei-Feng Huang, Jun Niu, and Qing-Huo Liu, Fellow, IEEE4

Abstract—In the framework of domain decomposition method,5

we present a novel hybrid solver for accurate electromagnetic6

simulations of composite objects in layered medium (LM). This7

hybrid solver combines the surface integral equation method and8

the ﬁnite element method to effectively minimize the simulation9

domain, where the effect of inhomogeneous background is10

involved by the LM Green’s functions, and the ﬁeld transmission11

between neighboring subdomains is realized by the Riemann12

solver. Numerical results are given to validate the proposed solver.13

Index Terms—Finite element method (FEM), layered medium,14

surface integral equation (SIE).15

I. INTRODUCTION16

OMAIN decomposition method (DDM) is one of the most17

popular research topics in electromagnetic engineering.18

DDM generally divides a complex domain into several smaller19

subdomains and solves them separately. DDM has been exten-20

sively investigatedforfree-spaceapplications, but its application21

in layered medium (LM) is rarely reported. The application of22

DDM is much more difﬁcult in LM than in homogeneous back-23

ground due to the involved complex background for the former.24

As a result, the DDM for LM applications is still a challenging25

topic.26

DDM was ﬁrst developed for the ﬁnite element method (FEM)27

[1] and the ﬁnite-difference time domain (FDTD) [2], and then28

extended to the surface integral equations (SIEs) [3]. DDM with29

FEM or FDTD is only efﬁcient in homogeneous background30

simulations, rather than in LM simulations. This is because an31

intrinsic disadvantage of FEM and FDTD is the large truncation32

domain for the simulation in inhomogeneous background, which33

leads to a heavy computational workload. SIE is more suitable34

for LM simulations since it only needs to discretize the target35

surface and the effect of background is involved by the layered36

medium Green’s functions (LMGFs) [4]. Unfortunately, SIE37

is only feasible for the homogeneous target. Therefore, a good38

option is to combine their advantages and avoid their drawbacks.39

Namely, FEM is utilized to simulate the interior domain, while40

Manuscript received February 16, 2016; accepted June 08, 2016. Date of pub-

lication; date of current version. This work was supported in part by the NSFC

under Grant 61301032, the Natural Science Foundation of Chongqing under

Grant cstc2013jcyjA40037, and the Scientiﬁc and Technological Research Pro-

gram of Chongqing Municipal Education Commission under Grant KJ130531.

Y. Ren is with the Chongqing University of Posts and Telecommunications,

Chongqing 40065, China (e-mail: renyi_cq@hotmail.com).

W.-F. Huang, J. Niu, and Q. Liu are with the Department of Electrical and

Computer Engineering, Duke University, Durham, NC 27705 USA (e-mail:

hwfbreeze@gmail.com; jn92@duke.edu; qhliu@duke.edu).

Color versions of one or more of the ﬁgures in this letter are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/LAWP.2016.2582149

SIE is utilized to simulate the surface domain. It should be 41

pointed out that we are aware of very few literature reports 42

on this topic in LM applications [5], [6], and all the existing 43

papers about the hybrid SIE/FEM only focus on the conformal 44

mesh, rather than DDM with nonconformal mesh. Thus, a more 45

attractive scheme is the hybrid FEM/SIE based on DDM with 46

nonconformal meshes. 47

This letter combines DDM with SIE and FEM to present 48

a hybrid solver for the accurate electromagnetic simulation of 49

composite objects in LM. In this hybrid solver, simulation do- 50

main is divided into one surface domain and several volume sub- 51

domains. The surface domain is modeled by SIE with LMGFs, 52

whereas the volume subdomains are modeled by FEM. The Rie- 53

mann solver boundary condition is utilized to connect the ﬁelds 54

on different subdomain interfaces [7], [8]. The resulting matrix 55

equation is solved by the Krylov subspace solver with precon- 56

ditioning method [9], [10]. Numerical results are then given to 57

validate the accuracy of this hybrid solver. 58

II. FORMULATION 59

Suppose a composite object with permittivityε(r) and perme- 60

ability μ(r) immersed in a planarly LM. The simulation domain 61

is decomposed into N

Sub

FEM subdomains and one SIE do- 62

main. In the ith interior FEM subdomain, the ﬁelds E

and 63

can be formulated into an equivalent variational problem, 64

i.e., 65

F (E







∇×E





∇×E



− k

·E



+ jk



∂ Ω



∂ Ω



· ˆndS,

i =1,...,N

Sub

(1)

where ˆn denotes the outward unit normal vector on ∂Ω

, and k

is the free-space wavenumber.

= Z

, and Z

is the free- 67

space intrinsic impedance. When the curl-conforming bases are 68

used to expand the ﬁelds, we have 69



m =1



m =1

(2)

where N

is the degrees of freedom (DOF) in Ω

, and {f

} 70

are the curl-conforming bases. E

and H

are the expansion 71

coefﬁcients of E

and

, respectively. A standard Galerkin 72

See http://www.ieee.org/publications

standards/publications/rights/index.html for more information.

IEEE Proof

2 IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 00, NO. 00, 2016

weak-form discretization of (1) yields73





(∇×f

) ·



∇×E



− k

· E



− jk



∂ Ω



∂ Ω



· ˆndS = S

i =1,...,N

Sub

,m=1,...,N

(3)

where S

is the excitation in ith subdomain. Then, the FEM74

matrix can be readily obtained from (3).75

The surface domain ∂Ω, which is connected with the back-76

ground, can be modeled by SIE. The tangential ﬁelds E

SIE

and77

SIE

on ∂Ω are connected with the equivalent electric cur-78

rents J

SIE

and magnetic currents M

SIE

by J

SIE

=ˆn × H

SIE

and M

SIE

= E

SIE

× ˆn, respectively. M

SIE

and J

SIE

can be80

expanded by the divergence-conformal bases g, i.e.,81

SIE

∂ Ω



m =1

SIE

∂ Ω



m =1

SIE

(4)

where N

∂ Ω

is the DOFs on ∂Ω. With the electric ﬁeld integral82

equation, the scattered ﬁeld E

sca

outside Ω is expressed as83

sca

(r)=L[J

SIE

]+K[M

SIE

] (5)

where Lis electric-type source operator, and Kis magnetic-type

source operator, respectively. In LM, the operators Land K are85

deﬁned as86

L[J

SIE

]= −jω



∂ Ω



(r, r



)μ



) ·J

SIE



) (6)

K[M

SIE

]= −

(r)



∂ Ω



∇×

(r, r



)ε



) ·M

SIE



)

(7)

where ω is the angular frequency. ε

and μ

are the permit-87

tivity and permeability of background, respectively.

is the88

electric-type dyadic Green’s function, and

is the magnetic-89

type dyadic Green’s function. r is the observation point, and r



is the source point. In this letter, the matrix-friendly LMGFs are91

adopted for their explicit matrix-form and the convenience for92

implementation. Since this letter focuses on the hybrid method93

but not LMGF, the explicit form of this LMGF can be found in94

[4]. By the Galerkin method, it is easy to establish the matrix95

equation of SIE and solve it by the method of moments (MoM).96

In DDM, the boundary condition should be imposed to en-97

sure the correct solution. Here, the frequency-domain Riemann98

solver is utilized as the transmission condition on the boundaries99

for its good convergence behavior [7], [8]. It can be formulated100

as101

(ˆn × E

∂ Ω

)(Y

+ Y

)=ˆn × (Z

∂ Ω

− ˆn ×

∂ Ω

+ Z

∂ Ω

+ˆn ×

∂ Ω

) (8)

(ˆn ×

∂ Ω

)(Z

+ Z

)=ˆn × (Z

∂ Ω

+ˆn ×E

∂ Ω

+ Z

∂ Ω

− ˆn ×E

∂ Ω

) (9)

where Z

, Z

and Y

, Y

are the corresponding impedances102

and admittances of the neighboring ith and jth subdomains.103

∂ Ω

and E

∂ Ω

are the ﬁelds on the surface of104

ith and jth subdomains, respectively. In (8) and (9), E

∂ Ω

and105

∂ Ω

are the ﬁelds at the interface, which are usually substituted 106

by the ﬁelds in i or jth subdomains in practice. In DDM, the 107

boundary ﬁelds on ∂Ω

are coupled with the ﬁelds in neigh- 108

boring subdomains, and the corresponding connection can be 109

established by substituting (9) into (3). Thus, with the identity 110

) · ˆn = −(ˆn ×

) · f

and assuming that the 111

magnetic ﬁeld in the left-hand side (LHS) of (9) is the same as 112

the magnetic ﬁeld in the ith domain, we have 113





∇×f



· μ

−1



∇×E



− k

· ε

· E



+ jk



∂ Ω

ˆn ·f

∂ Ω

+ Z



−Z

∂ Ω

+ˆn ×E

∂ Ω

+ Z

∂ Ω

− ˆn × E

∂ Ω



dS = S

i, j =1,...,N

Sub

,m =1,...,N

∂ Ω

(10)

where f

/∂Ω

is the testing basis on Ω

/∂Ω

. Equation (10) 114

cannot yield a square impedance matrix for each FEM subdo- 115

main. Thus, (8) is utilized to make the impedance matrix square. 116

This equation also connects the ﬁelds on ∂Ω

and on the surface 117

of neighboring subdomains. Assuming that the electric ﬁeld in 118

the LHS of (8) is the same as the electric ﬁeld in ith domain, we 119

then have 120



∂ Ω

· ˆn ×



−Y

∂ Ω

− ˆn ×

∂ Ω

+ Y

∂ Ω

+ˆn ×

∂ Ω



dS =0,

i =1,...,N

Sub

,m =1,...,N

(11)

For the similar reason, (8) and the same assumption will 121

be used again to obtain the submatrix of the SIE domain, and 122

connect the ﬁelds with FEM subdomains. Therefore, with the 123

Galerkin method, we have 124



∂ Ω

· ˆn ×



−Y

SIE

− ˆn × H

SIE

+ Y

∂ Ω

+ˆn ×H

∂ Ω



dS =0,

j =1,...,N

Sub

− 1,m =1,...,N

∂ Ω

(12)

According to the (5) and (10)–(12), we can obtain the ma- 125

trix equation of the whole system [7]. It is worth mentioning 126

that in the above formulas, the ﬁelds on the interfaces are usu- 127

ally expanded by nonconformal bases. Namely, different with 128

∂ Ω

and H

∂ Ω

, E

∂ Ω

and H

∂ Ω

will be expanded by the non- 129

conformal bases. Without loss of generality, the DDM matrix 130

equation of two subdomains, i.e., one SIE subdomain and one 131

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