Research On Radial Basis Function Meshless Method In Numercial Computation Of Electromagentic Field

The main methods for numerical computation engineering electromagnetic field are mesh methods which are based on mesh generation and local approximation. Therefore they meet two big challenges, one is the subdivision for complex structure, the other is the contradiction between precision and computation cost. Radial basis function (RBF), widely used in data interpolation and differential equation solving, is presently applied to form a collocation type meshless method for mathematical physical equations. So this thesis introduces the RBF to solve the above problems in the numerical computation of electromagnetic field by taking advantage of its high approximation precision and non-mesh generation. Then the principles, implementation schemes and discrete model of the RBF in solving typical engineering electromagnetic problems are investigated in a systematical manner.The RBF method's principles for interpolation and differential equation solution are presented firstly. Comparison study between RBF interpolation and traditional methods reveals that the RBF is characterized by precision, stability and implementation simplicity. Simultaneously, RBF can obtain continuous, smooth and precise solution only under a few nodes. On the other hand, the RBF shows obvious performance superiority in solving differential equation.In order to form a systematic RBF method for calculating electromagnetic field, the engineering projects are divided into static field model, time varying field model and eigen model according to their characteristics. Then each type of model is separately studied and following achievement is obtained.In chapter 3, the thesis constructed the RBF method for solving boundary value problem in static electromagnetic field. The research included: 1) On the basis of potential function boundary value model in static field, the RBF principles, implementation scheme and discrete model are presented, and then successful instances on electrostatic field, constant electric field and constant magnetic field are performed. 2) considering the RBF interpolation's approximate precision is relatively low near the boundary, the thesis carried out special research on boundary condition in boundary value model and then proposed such boundary treatment methods as follows: i. Owing to the existence of sudden change boundary in boundary I value model, the study introduced the compact support quasi Shannon wavelet function, and formed a coupling method which consists of RBF and the wavelet basis function. This method could combine the RBF's global approximation ability with the wavelet function's local approximation ability. ii. Owing to the existence of derivative boundary in the boundary II value model, the paper proposed a series of improvement technology as the floating point method, the Hermite collocation and the regular grid method which reflected the performance superiority of improved method. iii. Owing to the engagement boundary in multi-medium problems, the thesis proposed the subzone approximation scheme which obtained higher precision solution than global approximation and better manifested the influence of medium boundary.In chapter 4, the thesis formed the RBF methods for parabolic equation. Firslty, the time-varying field is classified into harmonic steady field and transient field, and then the phase form boundary model is the essence of harmonic steady field. Finally, the research focuses on the treatment of time variables in the transient fields. The studies included the following two methods --- the time-domain RBF method and the frequency domain RBF method. The former adopted the RBF approximating space function and the Crank-Nilcoson difference for time function. The latter is implemented through three stages. First, the time function is decomposed into complex exponential function by Fourier transform, then the solution in frequency domain is obtained through superposition of single frequency point time-harmonic field, and finally the solution in time domain is acquired by inverse Fourier transform. The result proved the RBF method can be successfully applied in transient eddy analysis.In chapter 5, the thesis put forword the RBF coupled method for electromagnetic eigen model analysis which is used to calculate the electromagnetic parameters, such as cut-off frequency, cut-off wave number and eigen functions. Compared with FEM method, the RBF gained higher accuracy and non-pseudo-solution phenomenon in the waveguide and resonator analysis. Furthermore, the RBF maintained high accuracy in high order model.In chapter 6, the RBF method was applied to the microstrip, the electrical alveolar , the transient field and irregular waveguide applications. A series of comparison between RBF and FEM were performed which were implemented with Maxwell 3D, Matlab PDE Toolbox. The results showed the RBF methods'validity in solving the electromagnetic instances.The systematic study on RBF showed that the RBF method is independent of mesh in solution procedure, so it avoids the trouble of subdivision in complex structure. Due to the good continuity, smoothness and boundless derivative advantages of global RBF, the RBF gains higher precision than local approximation methods. In addition, RBF is defined as the distance function, therefore it is insensitive to dimension and can be easily extended to high-dimensional analysis. Finally, since RBF method is flexible in setting the centers and collocation points, it is easy in implementation and programming. Considering its feasibility, effectiveness and outstanding performance advantages in the calculation of the electromagnetic, RBF will become an important numerical method in electromagnetic calculation.