^{1}

^{1}

^{1}

^{1}

^{1}

Hierarchical (

Hierarchical (

This paper is organized as follows. Section

We first proceed with a description of the IE method for solving electromagnetic scattering problems from 3D perfectly electric conductor (PEC). For concise introduction, only the electric field integral equation (EFIE) [

Discretize integral equation (

Considering that the construction of proposed multilevel SAI preconditioner is built upon the structure of

Each vertex

The electromagnetic interaction of any two clusters, including self-interaction, maps certain subblock of the system coefficient matrix. Practically, most of these subblocks can be approximated by low-rank matrices with high-accuracy. Therefore, a systemic and appropriate partitioning procedure is needed for the coefficient matrix. Based on the cluster tree

If two clusters are well separated geometrically, the Green function which connects the interaction of them is barely varying in their domains. That means, only a few patterns of interactional vectors can represent the whole mode; therefore, the subblock representing their interaction is rank deficit. In order to discerning these subblocks appropriately, we introduce the admissibility condition [

Two basis function clusters domains and their distance, describe the definition of

If the Krylov iterative methods are used to solve the linear system, we always expect to find a high-performance preconditioner to accelerate the convergence. Generally, instead of solving the linear system of the form

For a block-cluster tree

The intuitive grasp of

The data distribution of the finest 3 levels of

Level 1

Level 2

Level 3

Correspondingly, a set of preconditioning matrices

Because

If the system matrix is severely ill-conditioned and even the iterative solver with powerful preconditioner cannot obtain acceptable results, fast direct solvers are good alternative for IE

H_Inverse

The operator

For numerical implementation, hierarchical inversing is not as fast as hierarchical LU decomposition, which is another fast direct method based on matrices decomposition [

computing

computing

computng

computing

Process (

computing

computng

computing

Comparing with the procedure of hierarchical LU decomposition, one step is removed because we can obtain

To investigate the complexity of constructing IE

Computational complexity of constructing

Computational complexity of constructing

From the investigation, we can easily see that if incident wave frequency is fixed, the complexity of both CPU time and memory usage approach to

Firstly, a conducting sphere is used to demonstrate the improvement of the spectrum characteristics of the linear coefficient matrix by employing multilevel-SAI preconditioner (ML-SAI). Supposing

The regression index of different levels of ML-SAI and conventional SAI preconditioning.

Regression index | Number of unknowns | ||||

957 | 3,972 | 16,473 | 65,892 | ||

5.568 | 2.297 | 1.371 | 1.042 | ||

0.1311 | 0.2440 | 0.3155 | 0.4237 | ||

0.08925 | 0.2034 | 0.2523 | 0.2816 | ||

1.116 | 0.1112 | 0.1828 | 0.2165 | ||

ML-FMA | None | 5.723 | 2.415 | 1.355 | 0.989 |

SAI | 0.1052 | 0.2237 | 0.2844 | 0.3125 |

From Table

Next, a

The bistatic RCS of a conducting sphere with 40

The iterative history of solving the PEC sphere cases. GMRES(30) is used and accelerated by different preconditioners.

Another example is an aircraft model which is

The surface current distribution of the aircraft model obtained by solving the ML-SAI preconditioned ^{2}.

GMRES(90) is used to solve the linear system. The convergence histories of iteration with different preconditioners are presented in Figure

The time and memory cost of iterative solving aircraft model with different preconditioners.

Time (seconds) | Memory (MBytes) | |

ML-SAI level 1 | 582.8 | 325.4 |

ML-SAI level 2 | 234.5 | 464.5 |

ML-SAI level 3 | 184.3 | 786.5 |

SAI with MLFMA | 643.7 | 312.3 |

DBP | 1134.2 | 213.9 |

The convergence histories of GMRES(90) with no preconditioner, DB preconditioner, and ML-SAI preconditioners of different levels.

In this part, we give some numerical examples solved by fast

The bistatic RCS solved by hierarchical fast direct solvers. CFIE is solved by LU decomposition solver, and EFIE is solved by

Next, the computational complexity of fast

CPU time complexity of hierarchical

Memory usage complexity of hierarchical

The aircraft model shown in the previous part is also used here to test fast

The surface current distribution of the aircraft model obtained by solving EFIE with hierarchical ^{2}.

The most notable advantage of fast direct solver is the high efficiency of handling multiple right-hand-side cases. After LU or

The monostatic RCS of the aircraft model. The incident wave is 3.0 GHz, HH-polarization.

The hierarchical matrice methods presented in this paper is embedded in electromagnetic IE method. Due to its special structure, we can construct a multilevel SAI preconditioner to accelerate the convergence of iterative solving, and even kinds of fast direct solvers can be made, which is not viable for the traditional IE fast algorithm. The multilevel SAI preconditioner proposed here is more efficient than conventional “single level” preconditioners, and hierarchical fast direct solvers are good alternatives to iterative solvers, very suitable for ill-conditioned system and multiple right-hand-side problems. Furthermore, the kernel independence feature of hierarchical matrices method is adapted to varied electromagnetic problems without being limited to integral equation with free-space Green function.

This work is supported by the Fundamental Science Research Foundation of National Central University for Doctoral Program (E022050205) and partly supported by NSFC (no. 60971032), the Programme of Introducing Talents of Discipline to Universities under Grant b07046.