HighPerformance Computing and Engineering Applications in Electromagnetics
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MingLin Yang, XinQing Sheng, "On the Finite Element Tearing and Interconnecting Method for Scattering by Large 3D Inhomogeneous Targets", International Journal of Antennas and Propagation, vol. 2012, Article ID 898247, 6 pages, 2012. https://doi.org/10.1155/2012/898247
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 FETIDPH designed in [5], was applied to solve largescale phasedarray antenna and photonic crystal problems and developed as FETIDPEM1 in [6], where a global preconditioner is smartly designed to significantly improve the convergence of the interface solution in FETI. To maintain highspeed convergence of the interface solution for extremely largescale problems, the Robintype transmission conditions at the subdomain interfaces were employed to derive the interface equation instead of DirichlettoNeumann boundary condition in FETIDPEM1 in [7]. This latest algorithm of FETI in CEM is named as FETIDPEM2, which shows much better performance than FETIDPEM1, especially for extremely largescale problems. Recently, the FETIDPEM2 is extended to scattering problems [2]. However, few researchers give the detailed investigation on the performance of FETIDPEM2 for scattering by large inhomogeneous targets.
In this paper, the FETIDPEM1 and FETIDPEM2 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 FETIDPEM Formulation
The 3D scattering by a target can be formulated as where and are the freespace wave number and intrinsic impedance, respectively, and is an internal impressed current. represents a boundary, where the field satisfies the absorbing boundary condition (ABC) [10] with as boundary excitation.
According to the FETI method, the computational domain is divided into nonoverlapping subdomains (), where the superscript denotes the subdomain index. The interface shared by the two adjacent subdomain th and th is denoted as . The edges shared by more than two subdomains are called corner edges, denoted as . The edges shared by two subdomains and the surface of absorbing boundary are also treated as corner edges. The interior fields in each subdomain still satisfy (1). In the FETIDPEM1, the boundary condition of each subdomain is formulated by introducing as In the FETIDPEM2, the boundary condition of each subdomain is formulated by introducing as
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 where the subscripts , , and denote the unknowns associated with the internal volume, interfaces, and corner edges, respectively. The unknowns associated with the internal volume and interfaces are considered as local variables, whereas the unknowns associated with the corner edges are considered as global variables. The unknowns are eliminated from the FEM matrix equation for the th subdomain to obtain the corner equation of . Afterwards, the obtained corner equations for all subdomains are assembled together to get the following global corner equation: where , , can refer to [6]. Solving (8) to obtain and then substituting back to the FEM matrix equation for each subdomain, we can obtain the expression of in terms of . For FETIDPEM1, the continuity of tangential electric field between two adjacent subdomain is enforced to , and the final equation can be obtained where , , , and can refer to [6]. For FETIDPEM2, the continuities of tangential electric and magnetic fields between two adjacent subdomain are enforced to , and the final equation has the same form as (9) and has double unknowns compared with FETIDPEM1.
Since the inverse of in (9) essentially performs as a global preconditioner, (9) is usually well conditioned. Numerical experiments in [6, 7] are performed on a 2Dextended periodical structure radiation problem. Recently, the FETIDPEM2 is extended to 3D scattering problems [2]. However, few researchers give the detailed investigation on the performance of FETI for scattering by 3Dextended inhomogeneous targets, especially when the subdomain numbers are increased or material parameters are changed in all , , and 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 FETIDPEM 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 FETIDPEM methods. We compute the bistatic RCS of a metallic cube at 0.3 GHz with , . A tetrahedral mesh with average edge length of is used. In this calculation, the total FEM computation domain is terminated by enclosing the cube with a cubic outer boundary placed from the cube. Numerical results obtained are compared in Figure 1 with those obtained by the conventional FEM, and the MOMbased solution using the commercial software FEKO. Good agreement can be observed among these results.
To demonstrate the efficiency and capability of the two FETIDPEM methods, the following numerical experiments are performed on a dielectric brick. The absorbing boundary is placed away from the surface of the brick. The solution domain is divided into segments in direction and segments in direction as shown in Figure 2(a), which is called 2Dextended decomposition. The solution domain also can be divided into , , and segments in , , and directions, respectively, which is called 3Dextended decomposition.
(a) 2D extended
(b) 3D extended
First, let us investigate the performance of FETI for 2Dextended decomposition. The subdomain is a dielectric cube with side length having , . Mesh size is fixed as . We increase the frequency of the incident wave from 0.3 GHz to 0.6 GHz. The iteration numbers required by FETIDPEM1 and FETIDPEM2 with frequency and number of subdomains are presented in Figures 3(a) and 3(b). As we see, the iteration number required by FETIDPEM2 almost maintains constant with frequency and number of subdomains, whereas that by FETIDPEM1 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.
Second, let us investigate the performance of FETI for 3Dextended decomposition. We fix subdomain size as with incident frequency 0.3 GHz and increase , , and at the same time. Table 1 lists number of iterations for FETIDPEM1 and FETIDPEM2. It can be seen that the iteration number required by FETIDPEM2 increases with the number of subdomains. To further show the differences between different types of domain decomposition, we fix subdomain size as and change from 1 to 3. Figure 4 shows the iteration number of the FETIDPEM2 versus number of subdomains for different with different incident degree. It can be seen from Figure 4 that, for the 2Dextended domain decomposition, the iteration numbers maintain nearly constant with number of subdomains for different directions of incident wave, but for the 3Dextended 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 with and with . 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.

Third, we will investigate numerical performance of FETIDPEM2 versus inhomogeneity. Our numerical experiments are performed on a dielectric cube with at 0.3 GHz. We set two cases of the relative dielectric constant and , respectively. Mesh size is fixed as 0.05 m and 0.025 m for these two cases, respectively. The size of subdomain is , and 64 subdomains are used. The iteration numbers required by FETIDPEM2 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 FETIDPEM2, the parameter in (6) is chosen as . In fact, this parameter can be modified to get a better convergence. Table 2 shows that the iteration number required by FETIDPEM2 versus . It can be seen from Table 2 that the almost best convergence of FETIDPEM2 can be achieved by choosing , which agrees with that in [12].

To further investigate numerical performance of FETIDPEM2 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 , , , and , respectively. The thickness of each layer is . Figure 6 presents the iteration number required by FETIDPEM2 versus . It can be obtained from Figure 6 that the optimal performance of FETIDPEM2 is achieved by choosing , where can be approximated as average propagation constant of .
At last, to show the great capability of the FETIDPEM method, we compute the scattering by a large dielectric brick up to 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 . For lossless inhomogeneous case, the brick is filled with material of and repeatedly along direction with thickness of 0.5 m. For lossy inhomogeneous condition, the dielectric constants of material are set to and , respectively. The bistatic RCSs by these bricks are shown in Figure 7. In these calculations, subdomain size is , and the ABC is placed away from the brick. Total subdomains are used. Computation resources needed in this calculation are listed in Table 3.

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 FETIDPEM1 becomes seriously slow with the number of subdomains, especially for high frequency. FETIDPEM2 is much faster than FETIDPEM1 with number of subdomains. However, FETIDPEM2 also cannot maintain the convergence speed with number of subdomains for the 3Dextended decomposition. Furthermore, the convergence speed of FETIDPEM2 highly depends on inhomogeneity of targets. Taking inhomogeneity into the coefficient in the Robin transmission condition can improve the convergence speed of FETIDPEM2.
Acknowledgment
This work was supported by the NSFC under Grant 10832002.
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Copyright
Copyright © 2012 MingLin Yang and XinQing Sheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.