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.

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 [

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.

The 3D scattering by a target can be formulated as

According to the FETI method, the computational domain is divided into nonoverlapping subdomains

The fields in each subdomain can be determined under the boundary condition (

Since the inverse of

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 [

The first numerical experiment is to show the accuracy of the two FETI-DPEM methods. We compute the bistatic RCS of a

Bistatic RCS of the metallic cube with

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

Domain decomposition style of a brick domain.

2D extended

3D extended

First, let us investigate the performance of FETI for 2D-extended decomposition. The subdomain is a dielectric cube with side length

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

Iteration number versus frequency.

Iteration number versus number of subdomains.

Second, let us investigate the performance of FETI for 3D-extended decomposition. We fix subdomain size as

Iteration number versus the number of subdomains.

Number of subdomains | FETI-DPEM1 | FETI-DPEM2 | |||

Dual unknowns | Iteration number | Dual unknowns | Iteration number | ||

27 | 2160 | 37 | 4320 | 40 | |

125 | 12000 | 359 | 24000 | 69 | |

1000 | 108000 | >500 | 21600 | 127 |

Iteration number versus number of subdomains for different

Third, we will investigate numerical performance of FETI-DPEM2 versus inhomogeneity. Our numerical experiments are performed on a dielectric cube with

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

1.5 | 2 | 2.5 | 3 | 3.5 | ||
---|---|---|---|---|---|---|

105 | 87 | 84 | 86 | 93 | 103 | |

321 | 254 | 233 | 228 | 232 | 250 |

To further investigate numerical performance of FETI-DPEM2 for inhomogeneity, we perform an experiment for inhomogeneous brick as shown in Figure

An inhomogeneous dielectric brick with four different materials.

Iteration number for the brick versus different

At last, to show the great capability of the FETI-DPEM method, we compute the scattering by a large dielectric brick up to

Computation resources for the large dielectric brick.

Material type | Dual unknowns (M) | Corner unknowns (M) | Total unknowns (M) | Memory | Iteration Number | Total CPU time (Min) |
---|---|---|---|---|---|---|

Homogeneous | 215 | 323 | ||||

Inhomogeneous (Lossless) | 9.2 | 0.2 | 41 | 8.2 | 388 | 481 |

Inhomogeneous (Lossy) | 54 | 105 |

Bistatic RCS of the large dielectric brick with

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.

This work was supported by the NSFC under Grant 10832002.