Research On The Fast Algorithms Based On The Higher Order Hierarchical Vector Basis Functions

The electromagnetic scattering is very important in many fields such as modern radar target recognizing, microwave imaging and microwave remote sensing. In order to meet the practical engineering requirement, the field of computational electromagnetics has seen a considerable surge in research on efficient accurate numerical methods. The emphases of this paper are concentrating on the application of the higher order hierarchical vector basis functions in the electromagnetic integral equation method, specially majored on the methods to improve the computational property when those bases are used to get the electromagnetic response of the target. This paper is composed as the following four parts.The first part is the application of those bases in the frequency domain integral equation. The near orthogonal basis functions are introduced at first, which based on the modified Legendre polynomials. Then, the influences of the orthogonality on the bases have been investigated and the result is that the bases will have a well computational property when they have a better orthogonality. So the following part will emphasize the orthogonality and analyze the maximally orthogonalized higher order vector basis functions. The scaling factor was reformed to speed up the iteration convergence in the numerical solution. Furthermore, the radiating far field of the higher order bases which defined on the large patches has been analyzed. The result is that a new method to sparsify the impedance matrix and relief the memory pressure is introduced when the higher order vector basis functions defined on large patches have been utilized in the numerical solution of integral equations. At last, the idea of near orthogonal basis functions has been extended into the triangular case as a new higher order hierarchical vector bases defined on the triangular elements is introduced.The second part is the application of the near orthogonal hierarchical vector bases in the time domain integral equation method (TDIE). In this part, the algorithm of the near orthogonal bases in the TDIE is investigated, and the numerical example in this part will show the well property of this method. Then, after briefly discussing several causes of the late-time instability of the TDIE solvers, a novel viewpoint about the instability is proposed. A new method has been invented to calculate the element of the impedance matrix accurately in TDIE. This method is adapted to the higher order geometric model which will lead it universal in TDIE. At last, some work on the temporal response of the moving PEC target by TDIE is introduced. Furthermore, the temporal response of the multi-target with a relative movement is investigated too, which will use the domain decomposition method in time domain.The interpolation/extrapolation method of the information in time and frequency domain researched in the third part. The difficulties in the time domain method will be listed in this part at first. Then the interpolation/extrapolation method will be introduced as it is a perfect method to delay the instability of explicit MOT algorithm. This method will use the low frequency information in the frequency domain to complement the information of the latter temporal response of the target which will deal the instability. An adaptive sampling method is introduced when the principle of the interpolation/extrapolation method have been investigated. Then, the near orthogonal hierarchical vector basis functions are used to get the sampling information in the frequency and time domain respectively, which will improve the efficiency in frequency domain dramatically. At last, the physical bases of the interpolation/extrapolation method have been emphasized and a new method to consolidate the results by the interpolation/extrapolation method is invented.The fourth part has deal with the problem of the large size patch is hard to model the surface of the complex structure. This problem is the foremost difficulty in utilizing the higher order bases to solve the engineering problem. The method introduced in this paper will make it flexible in geometry model by separate the surface of the target into slick and subtle areas. Then the different size patches are used to mesh the different areas. This method will make sure the continuity of the normal current along the common edge between the different areas and get a flexible meshing method. Furthermore, a new adaptive meshing method in introduced as the extend of this meshing method.The studies of this paper demonstrate the high accuracy and efficiency of higher order hierarchical vector bases in the integral equation method. The analysis and the numerical results in this paper display its potential to solutions of electromagnetic engineering problems.