On the Finite Element Tearing and Interconnecting Method for Scattering by Large 3D Inhomogeneous Targets

Abstract

The finite element tearing and interconnecting method (FETI) is applied to compute scattering by large 3D inhomogeneous targets. Two algorithms of FETI have been implemented for 3D scattering. The performance of these two FETI algorithms has been investigated in detail, particularly for large inhomogeneous targets. Numerical experiments show that the performance of FETI relies on the style of domain decomposition and inhomogeneity, which has not been carefully studied before. A trick for improving convergence of FETI is presented for inhomogeneous targets.

1. Introduction

The domain decomposition has been recognized as one of most important methodologies for constructing efficient parallel computing algorithms in recent years. Among various domain decomposition methods of the finite element method (FEM), the finite element tearing and interconnecting (FETI) shows great potential to improve the capability of the finite element method [1–9]. The FETI designed in [3] was applied to the electromagnetic field problems in [4]. The idea of a more efficient version of FETI, named as FETI-DPH designed in [5], was applied to solve large-scale phased-array antenna and photonic crystal problems and developed as FETI-DPEM1 in [6], where a global preconditioner is smartly designed to significantly improve the convergence of the interface solution in FETI. To maintain high-speed convergence of the interface solution for extremely large-scale problems, the Robin-type transmission conditions at the subdomain interfaces were employed to derive the interface equation instead of Dirichlet-to-Neumann boundary condition in FETI-DPEM1 in [7]. This latest algorithm of FETI in CEM is named as FETI-DPEM2, which shows much better performance than FETI-DPEM1, especially for extremely large-scale problems. Recently, the FETI-DPEM2 is extended to scattering problems [2]. However, few researchers give the detailed investigation on the performance of FETI-DPEM2 for scattering by large inhomogeneous targets.

In this paper, the FETI-DPEM1 and FETI-DPEM2 are implemented for 3D scattering by large inhomogeneous targets. The performance of these two FETI algorithms has been investigated in detail, particularly for large highly inhomogeneous targets and different domain decomposition types. An implementation trick of FETI is presented for inhomogeneous targets.

2. The FETI-DPEM Formulation

The 3D scattering by a target can be formulated as 1∇×𝜇_𝑟∇×𝐸−𝑘₀𝜀_𝑟𝐸=−𝑗𝑘₀𝑍₀𝐽_imp,inΩ∈𝑅³,𝑛×𝐸=0,on𝜕Ω_PEC,𝑛×∇×𝐸=0,on𝜕Ω_PMC,𝑛×∇×𝐸+𝑗𝑘₀𝑛×𝑛×𝐸=𝑈,on𝜕Ω_ABC, (1)(2)(3)(4) where 𝑘₀ and 𝑍₀ are the free-space wave number and intrinsic impedance, respectively, and 𝐽_imp is an internal impressed current. 𝜕Ω_ABC represents a boundary, where the field satisfies the absorbing boundary condition (ABC) [10] with 𝑈=𝑛×∇×𝐸_inc+𝑗𝑘₀𝑛×𝑛×𝐸_inc as boundary excitation.

According to the FETI method, the computational domain is divided into nonoverlapping subdomains Ω^𝑝 (𝑝=1,2,…,𝑁_𝑝), where the superscript denotes the subdomain index. The interface shared by the two adjacent subdomain 𝑖th and 𝑗th is denoted as Γ_𝑖𝑗. The edges shared by more than two subdomains are called corner edges, denoted as Γ_𝑐. The edges shared by two subdomains and the surface of absorbing boundary are also treated as corner edges. The interior fields in each sub-domain still satisfy (1). In the FETI-DPEM1, the boundary condition of each subdomain is formulated by introducing Λ as 𝑛^𝑝×1𝜇^𝑝_𝑟∇×𝐸^𝑝=−𝑛^𝑞×1𝜇^𝑞_𝑟∇×𝐸^𝑞=Λ. (5) In the FETI-DPEM2, the boundary condition of each subdomain is formulated by introducing Λ as 𝑛^𝑝×1𝜇^𝑝_𝑟∇×𝐸^𝑝+𝛼^𝑝𝑛^𝑝×𝑛^𝑝×𝐸^𝑝=Λ^𝑝. (6)

The fields in each subdomain can be determined under the boundary condition (5) or (6) by using the standard FEM. A special procedure is designed in FETI to solve the fields in each subdomain. The unknowns of the electric field in each subdomain are grouped into three categories 𝐸^𝑝=𝐸^𝑝_𝑉𝐸^𝑝_𝐼𝐸^𝑝_𝑐=𝐸^𝑝_𝑟𝐸^𝑝_𝑐, (7) where the subscripts 𝑉, 𝐼, and 𝑐 denote the unknowns associated with the internal volume, interfaces, and corner edges, respectively. The unknowns 𝐸^𝑝_𝑟 associated with the internal volume and interfaces are considered as local variables, whereas the unknowns 𝐸^𝑝_𝑐 associated with the corner edges are considered as global variables. The unknowns 𝐸^𝑝_𝑟 are eliminated from the FEM matrix equation for the 𝑝th subdomain to obtain the corner equation of 𝐸^𝑝_𝑐. Afterwards, the obtained corner equations for all sub-domains are assembled together to get the following global corner equation:𝐾_𝑐𝑐𝐸_𝑐=𝑓_𝑐−𝐾^𝑇_𝑟𝑐𝜆, (8) where 𝐾_𝑐𝑐, 𝐾^𝑇_𝑟𝑐, 𝑓_𝑐 can refer to [6]. Solving (8) to obtain 𝐸_𝑐 and then substituting 𝐸_𝑐 back to the FEM matrix equation for each subdomain, we can obtain the expression of 𝐸^𝑝_𝑟 in terms of 𝜆. For FETI-DPEM1, the continuity of tangential electric field between two adjacent subdomain is enforced to 𝐸^𝑝_𝐼, and the final equation can be obtained 𝐾_𝑟𝑟+𝐾_𝑟𝑐𝐾⁻¹_𝑐𝑐𝐾_𝑐𝑟𝜆=𝑓_𝑟−𝐾_𝑟𝑐𝐾⁻¹_𝑐𝑐𝑓_𝑐, (9) where 𝐾_𝑟𝑟, 𝐾_𝑟𝑐, 𝐾_𝑐𝑟, and 𝑓_𝑟 can refer to [6]. For FETI-DPEM2, the continuities of tangential electric and magnetic fields between two adjacent subdomain are enforced to 𝐸^𝑝_𝐼, and the final equation has the same form as (9) and has double unknowns compared with FETI-DPEM1.

Since the inverse of 𝐾_𝑐𝑐 in (9) essentially performs as a global preconditioner, (9) is usually well conditioned. Numerical experiments in [6, 7] are performed on a 2D-extended periodical structure radiation problem. Recently, the FETI-DPEM2 is extended to 3D scattering problems [2]. However, few researchers give the detailed investigation on the performance of FETI for scattering by 3D-extended inhomogeneous targets, especially when the subdomain numbers are increased or material parameters are changed in all 𝑥, 𝑦, and 𝑧 directions. We will give the detail investigation by numerical experiments in the next section on the performance of FETI for scattering by large inhomogeneous targets.

3. Numerical Experiments

To demonstrate the accuracy, efficiency, and capability of the two FETI-DPEM methods, a series of numerical experiments are performed in this section. All the computations are performed on a computer having 2 Intel X5650 2.66 GHz CPUs with 6 cores for each CPU, 32 GB memory. The GMRES solver in [11] is employed to solve (9) with a restart number of 20, and the convergence criterion is set to 0.005.

The first numerical experiment is to show the accuracy of the two FETI-DPEM methods. We compute the bistatic RCS of a 1m×1m×1m metallic cube at 0.3 GHz with 𝜃_inc=0^∘, 𝜑_inc=0^∘. A tetrahedral mesh with average edge length of 0.05m is used. In this calculation, the total FEM computation domain is terminated by enclosing the cube with a cubic outer boundary placed 0.5m from the cube. Numerical results obtained are compared in Figure 1 with those obtained by the conventional FEM, and the MOM-based solution using the commercial software FEKO. Good agreement can be observed among these results.

Figure 1 

Bistatic RCS of the metallic cube with 𝜃inc=0∘,  𝜑inc=0∘ at 0.3 GHz.

To demonstrate the efficiency and capability of the two FETI-DPEM methods, the following numerical experiments are performed on a dielectric brick. The absorbing boundary is placed 0.5𝜆₀ away from the surface of the brick. The solution domain is divided into 𝑀 segments in 𝑥-direction and 𝑁 segments in 𝑦-direction as shown in Figure 2(a), which is called 2D-extended decomposition. The solution domain also can be divided into 𝑀, 𝑁, and 𝐿 segments in 𝑥, 𝑦, and 𝑧 directions, respectively, which is called 3D-extended decomposition.

(a) 2D extended

(b) 3D extended

Figure 2 

Domain decomposition style of a brick domain.

First, let us investigate the performance of FETI for 2D-extended decomposition. The subdomain is a dielectric cube with side length 0.5m having 𝜀_𝑟=4, 𝑢_𝑟=1. Mesh size is fixed as 0.05m. We increase the frequency of the incident wave from 0.3 GHz to 0.6 GHz. The iteration numbers required by FETI-DPEM1 and FETI-DPEM2 with frequency and number of subdomains are presented in Figures 3(a) and 3(b). As we see, the iteration number required by FETI-DPEM2 almost maintains constant with frequency and number of subdomains, whereas that by FETI-DPEM1 increase with frequency and number of subdomain, especially for high frequency, which completely agree with the conclusions in [6, 7].

(a) Iteration number versus frequency.

(b) Iteration number versus number of subdomains.

Figure 3 

Iteration number required by FETI-DPEM1 and FETI-DPEM2.

Second, let us investigate the performance of FETI for 3D-extended decomposition. We fix subdomain size as 0.2m×0.2m×0.2m with incident frequency 0.3 GHz and increase 𝑀, 𝑁, and 𝐿 at the same time. Table 1 lists number of iterations for FETI-DPEM1 and FETI-DPEM2. It can be seen that the iteration number required by FETI-DPEM2 increases with the number of sub-domains. To further show the differences between different types of domain decomposition, we fix subdomain size as 0.5m×0.5m×0.5m and change 𝐿 from 1 to 3. Figure 4 shows the iteration number of the FETI-DPEM2 versus number of subdomains for different 𝐿 with different incident degree. It can be seen from Figure 4 that, for the 2D-extended domain decomposition, the iteration numbers maintain nearly constant with number of subdomains for different directions of incident wave, but for the 3D-extended domain decomposition, the iteration numbers are not constant again and increase with the number of subdomains. Furthermore, the increase speed of the iteration number with the number of subdomains changes for different incident wave directions. To exclude the possibility due to the increase in number of subdomains, another comparison is provided between 10×10 with 𝐿=3 and 30×10with 𝐿=1. The total subdomains are the same, but the numbers of iterations are 58 and 25, respectively. The latter is nearly the same as those in Figure 4.

Table 1 

Iteration number versus the number of subdomains.

Figure 4 

Iteration number versus number of subdomains for different 𝐿.

Third, we will investigate numerical performance of FETI-DPEM2 versus inhomogeneity. Our numerical experiments are performed on a dielectric cube with 1m×1m×1m at 0.3 GHz. We set two cases of the relative dielectric constant 𝜀_𝑟=4 and 𝜀_𝑟=9, respectively. Mesh size is fixed as 0.05 m and 0.025 m for these two cases, respectively. The size of sub-domain is 0.5m×0.5m×0.5m, and 64 sub-domains are used. The iteration numbers required by FETI-DPEM2 are 105 and 321 for two cases respectively. It shows that the performance of FETI highly depends on the inhomogeneity of the solution domain. It is worth to point out that in all above computation in FETI-DPEM2, the parameter 𝛼 in (6) is chosen as 𝑗𝑘₀. In fact, this parameter can be modified to get a better convergence. Table 2 shows that the iteration number required by FETI-DPEM2 versus 𝛼. It can be seen from Table 2 that the almost best convergence of FETI-DPEM2 can be achieved by choosing 𝛼=𝑗𝑘=𝑗√𝜇_𝑟𝜇₀𝜀_𝑟𝜀₀, which agrees with that in [12].

Table 2 

Iteration number required by FETI-DPEM2 for dielectric cube with different materials versus 𝛼.

To further investigate numerical performance of FETI-DPEM2 for inhomogeneity, we perform an experiment for inhomogeneous brick as shown in Figure 5. This brick has four layers, which have different dielectric constants with 𝜀₁=2, 𝜀₂=3, 𝜀₃=4, and 𝜀₄=5, respectively. The thickness of each layer is 0.5m. Figure 6 presents the iteration number required by FETI-DPEM2 versus 𝛼. It can be obtained from Figure 6 that the optimal performance of FETI-DPEM2 is achieved by choosing 𝛼=𝑗𝑘, where 𝑘 can be approximated as average propagation constant of 𝑘=(𝑘₁+𝑘₂+𝑘₃+𝑘₄)/4=1.84𝑘₀.

Figure 5 

An inhomogeneous dielectric brick with four different materials.

Figure 6 

Iteration number for the brick versus different 𝛼.

At last, to show the great capability of the FETI-DPEM method, we compute the scattering by a large dielectric brick up to 10m×10m×5m at 0.3 GHz. Three types of material constitutions of the brick are taken into consideration, which are homogeneous, lossless inhomogeneous, and lossy inhomogeneous. For homogeneous case, the brick is filled with material of 𝜀_𝑟=2. For lossless inhomogeneous case, the brick is filled with material of 𝜀_𝑟=2 and 𝜀_𝑟=4 repeatedly along −𝑧 direction with thickness of 0.5 m. For lossy inhomogeneous condition, the dielectric constants of material are set to 𝜀_𝑟=2 and 𝜀_𝑟=4−𝑗, respectively. The bistatic RCSs by these bricks are shown in Figure 7. In these calculations, subdomain size is 0.5m×0.5m×0.5m, and the ABC is placed 0.5m away from the brick. Total 22×22×12 subdomains are used. Computation resources needed in this calculation are listed in Table 3.

Table 3 

Computation resources for the large dielectric brick.

Figure 7 

Bistatic RCS of the large dielectric brick with 𝜃inc=0∘,  𝜑inc=0∘ at 0.3 GHz.

4. Conclusions

The FETI with the absorbing boundary condition (ABC) is applied to electromagnetic scattering by large inhomogeneous targets in this paper. The convergence speed of FETI-DPEM1 becomes seriously slow with the number of subdomains, especially for high frequency. FETI-DPEM2 is much faster than FETI-DPEM1 with number of subdomains. However, FETI-DPEM2 also cannot maintain the convergence speed with number of sub-domains for the 3D-extended decomposition. Furthermore, the convergence speed of FETI-DPEM2 highly depends on inhomogeneity of targets. Taking inhomogeneity into the coefficient in the Robin transmission condition can improve the convergence speed of FETI-DPEM2.

Acknowledgment

This work was supported by the NSFC under Grant 10832002.