Research On Meshless Method With Radial Basis Functions In Computational Electromagnetics

Meshless method (MLM) is a novel numerical method that bases on the nodes, not meshes. It is one of the most active front fields of numerical methods. In this paper, we take the MLM based on radial basis functions (RBF) collocation and point matching method (RBF-MLM) to solve the differential and integral equations of three kinds of electromagnetic (EM) problems—static field, time-harmonic field and transient field problems, and study the various key techniques of their computing processes. A numerical algorithm system of EM that bases on RBF-MLM is initially established. Our work expands the area of computational electromagnetics (CEM).Firstly, we introduce two kinds of RBF interpolations—direct method and indirect method. The RBF-MLM of general operator equations is deduced from the method of weighted residuals. Through studying the properties of RBF and its constructing shape function at the singular collocation nodes, we analyze the computation features of RBF-MLM solving the various differential and integral equations in EM problems.In the static field problems, we deduce the discretization formulation of the Poisson equation by RBF-MLM. With a metal groove example, we analyze the law and the various influence factors of RBF-MLM in detail. The direct and indirect methods of RBF-MLM are equivalent and"preconditioning"related according to thier constructed matrix properties. The calculation accuracy and efficiency can be obviously increased with the conformal node or multi-scale node techniques. For the drawbacks of the big matrix condition number and low efficiency by RBF-MLM, we introduce the domain decomposition method (DDM) of RBF-MLM to overcome them. The electrical charge integral equations (IE) are discretized by RBF-MLM, which are used to solve the one-dimensional (1-D) charged conductor line and the two-dimensional (2-D) charged conductor plate problems.In the time-harmonic field problems, we deduce the discretization formulation of 2-D Helmholtz equation in waveguide problems by RBF-MLM. With a rectangular waveguide example, we analyze the law and the various influence factors of RBF-MLM in detail. Through computing several waveguide examples with curve boundaries and singular boundaries, it is show that RBF-MLM is a general and brief numerical method to compute waveguide problems. The modes and three-dimensional (3-D) Helmholtz equation of the spherical coordinates in special sphere resonant cavity are simply processed; RBF-MLM in rectangular coordinates can be used to compute effectively the various degenerate modes in cavity. We also deduce the discretization formulations of The Hallen's equation, Pocklington's equation, magnetic field IE (MFIE) and electric field IE (EFIE) of EM radiation and scatter problems by RBF-MLM. The singular integral of impedance matrix element with RBF is effectively processed. Through computing the various examples of EM radiation and scatter problems, we analyze the law and the various influence factors of RBF-MLM in detail.In the transient field problems, we deduce the discretization formulation of the time-domain wave equation and 1-D and 2-D time-domain Maxwell equations by RBF-MLM, and deeply analyze the key techniques of the implementation, such as numerical stability, node dispersion errors, truncated boundary conditions and excitation source, etc. Thus, we obtain the relationship of various control parameters and a complete simulating procedure of RBF-MLM. The node dispersion of RBF-MLM has the characteristic of isotropy dispersion, so the field excited with the point source propagates as concentric wave. Through computing various examples of 1-D and 2-D transient field problems, we verify the process of numerical algorithm for RBF-MLM.