The Electromagnetics Domain Decomposition Time Domain Methods Based On Laguerre Polynomials

Weighted Laguerre polynomials as the time-domain basis functions express thetime-domain electromagnetic waveform effectively. Laguerre polynomials are appliedto time-domain electromagnetic methods. The time derivatives are handled analytically,at the same time, the time variable is eliminated from the computation, and the implicitand unconditionally stable time-domain method is obtained. So far, the time-domainmethods based on Laguerre polynomials are mainly the unconditionally stablefinite-difference time-domain (Laguerre-FDTD) method, unconditionally stablefinite-element time-domain (Laguerre-FETD) method, and unconditionally stabletime-domain integral equation (Laguerre-TDIE) method. The first two methods arebased on the differential equations, and the latter is based on the electromagneticintegral equations. In this thesis, we focus on the Laguerre-FDTD method andLaguerre-FETD method. On the basis of previous studies, the recursive formula ofLaguerre-FETD is improved and an efficient formula is obtained. We also propose anefficient domain decomposition (DD) Laguerre-FDTD method. The main contentincludes:1. The Laguerre polynomials and the Laguerre-FDTD method are introduced. Weemploy two numerical examples to illustrate the solution of electromagnetic problemsand the efficiency for electromagnetic problems with fine structures using theLaguerre-FDTD method, respectively.2. The process of FETD method and the unconditionally stable FETD methodusing New-mark-β method are presented, and the Laguerre-FETD method is introduced.For the conventional Laguerre-FETD, all unknowns of the orders need to be kept ateach order. The proposed recursive formula of Laguerre-FETD only needs to save theunknows of the current order, and reduces the complexity of the algorithm from squaredegree to linear degree. The numerical examples verify the efficiency of the proposedrecursive formula. The Laguerre-FETD method is applied to the solution of eigenvalueproblems. The numerical example of a circle dielectric-loaded waveguide shows itsadvantages of accuracy and efficiency compared with the conventional FETD method and Laguerre-FDTD method.3. Combining the implicit sparse relation of the unknowns for the Laguerre-FDTDmethod and matrix decomposition technique, we propose a novel domaindecomposition Laguerre-FDTD method. The interface of the different subdomains isnon-overlapping, and the coupling information is stored in the linear system generatedby the interface. Once the Schur somplement system of the interface’s linear system hasbeen solved, one can solve the linear systems generated by subdomains independently.So the proposed method is a natural parallel method. The DD-Laguerre-FDTD methodis used to simulate the radiation of a linear current source in the two-dimensioanl (2-D)TM case to verify the accuracy.The proposed method is applied to solve the2-D scattering problems. With thetotal-field/scattered-field boundary and Mur’s second order absorbing boundarycondition, the radar cross sections of two2-D structures are calculated. The numericalexamples verify the accuracy and efficiency of the proposed method.