Study And Application On Higher Order Vector FEM In Electromagnetic Field

The finite-element method (FEM) has become a very important tool in solving electromagnetic (EM) wave problems because of its ability to model geometrically and compositionally complex problems. How to construct element bases to get higher accuracy and faster convergence is a very important topic of active research. The emphases of this paper are concentrating on investigating of higher order vector finite element methods and its applications to three dimensional electromagnetic eigenvalue and scattering problems. This paper is composed as the following four parts.In the first part, the construction, calculation and performance comparison of different higher order interpolatory and hierarchical vector elements are researched systematically. Nedelec's functional spaces of H1(curl) tetrahedral vector elements are analyzed systematically by using the method combined Nedelec constraints and complete polynomials. The relations between various vector elements and the functional spaces are validated. By using the classified bases and block matrix technique, the explicit forms of the elemental matrices for the higher order vector elements are presented completely, and the explicit results of integration matrices are also given. The results of a numerical experiment that investigates the resonant problem of a rectangular cavity, compare the performance of different vector elements systematically (such as calculated accuracy, condition numbers, selectivity of the facet related basis functions). The methods can be extended to the analysis of ultra higher order vector elements with any type effectively.In the second part, the implementation for higher order interpolatory, hierarchical and curvilinear vector elements and some key issues are researched systematically. A new method is proposed to implement higher order interpolatory vector elements by using the classified bases technique, which can be applied to 3D vector elements with any type effectively. Based on the fact that hierarchical higher order basis functions permit elements of different orders to be used together in the same mesh, an efficient selective field expansion technology is studied, and a new method is proposed to implement mixed order hierarchical vector elements. The implementation for higher order curvilinear vector elements is researched systematically, and some key issues are discussed.In the third part, sparse matrix storage schemes and fast solve technology are researched systematically. Based on the fact that FEM matrix is sparse, the higher order vector FEM matrix is validated to have a random sparse structure, which is more efficient in storage in the case of varying bandwidth. Then, the bandwidth of the global matrix is reduced by RCM reordering technology. And some fast (direct, iterative) solving and preconditioned methods based on sparse technology are studied.In the fourth part, large-scale electromagnetic eigenvalue and scattering problems are analyzed by using H1(curl) tetrahedral vector elements. The results of a numerical experiment that investigates the resonant problem of a rectangular cavity, show the performance of ARPACK eigensolvers based on various solving techniques, which are also applied to the eigen-solution of complex cavities. 3D electromagnetic scattering problems are also analyzed by using H1(curl) tetrahedral vector elements and Webb-Kaneliopoulos vector absorbing boundary conditions.The studies of this paper demonstrate high accuracy and efficiency of higher order vector finite element method. The analysis and the numerical results display its tremendous potential to the solution of electromagnetic engineering problems.