Higher-Order Hierarchical Vector Basis Functions On Bézier Curved Triangular Surface And Their Applications In The Method Of Moments

The method of moments is one of the important methods for the analysis of electromagnetic problems. Since RWG basis functions based on triangle subdivision were proposed, the tradi-tional method of moments has been developed rapidly. However, RWG basis functions are of low order, and hence, a lot of unknowns will be generated for electrically large problems, re-sulting in high computational complexity and storage complexity. Many fast algorithms based on the method of moments have been developed in order to overcome the shortcomings of the traditional method of moments in computational complexity and storage complexity. Another method to reduce the computational complexity and storage complexity is the introduction of higher-order basis functions and the establishment of the higher-order moment method model. In this way, the division size can become larger, leading to the reduction of unknowns. Under the premise of the accuracy is guaranteed, the ability to solve the problem of electrically large size can be improved. At present, the research on higher-order method of moments has become a hot topic in Computational Electromagnetics. In this paper, we study the structure of higher-order hierarchical vector basis functions on Bézier curved triangular surface, and establish a high-order moment method model based on the triangulation with Bézier curved triangular surfaces as cells, and further establish an acceleration scheme based on the adaptive cross approximation algorithm. The main works of this thesis are as follows:1. The mathematical foundation of higher-order basis functions is described, binary orthog-onal polynomials defined on a flat right triangle are used to construct binary orthogonal polynomials on a general Bézier curved triangle, and hence, the theoretical foundation of higher-order basis functions on Bézier curved triangular surface is laid down.2. One type of higher-order hierarchical vector basis functions on Bézier curved triangular surface is proposed. These basis functions have the quasi-orthogonality, and can accu-rately describe the equivalent surface current, thus generating a well-conditioned matrix. Numerical examples demonstrate the correctness of the higher-order method of moments here.3. The higher-order moment method model using the higher-order basis functions here is established, and both the formula for calculating surface-divergence of the higher-order basis functions and the formula for processing the singular integrals are derived. Nu-merical examples confirm the correctness and robustness of the higher-order method of moments.4. The ACA algorithm is introduced into the higher-order method of moments based on Bézier triangulation. The near-field and far-field interactions are divided by means of the octree structure. The interaction between two well-separated basis function groups is compressed by using the ACA technique, and is stored in a factorial form, both reducing the matrix storage and accelerating the calculation of matrix-vector product. Numerical examples confirm both the feasibility of this acceleration scheme and the acceleration effect.