Study On Some Key Technique Of Two Dimensional DGTD Method

The Discontinuous Galerkin Time Domain(DGTD)technique is developed based on Finite Volume Time Domain,which has advantages of mesh flexibility from Finite Element Time Domain(FETD)and explicit iterative characteristics from Finite Difference Time Domain(FDTD).The wide application prospect of multi-scale problem and waveguide discontinuity problem etc.makes this algorithm become one of the hotspots in the area of electromagnetism in recent years.In this paper,some key problems of the DGTD numerical method in two dimensional transverse wave(TM)wave case are studied,including obtaining of DGTD iterative calculation formula,data exchange of numerical flux between units,the realization of absorption boundary conditions in DGTD,introduction of plane wave and extrapolation of near-field field and so on.In addition,the application of high order hierarchical basis function in DGTD algorithm is studied deeply.A DGTD treatment scheme for three kind of common dispersive media models is also presented,and this scheme is applied to the analysis of electromagnetic problems related to dusty plasma.The specific research contents and main innovations are as follows:(1)The concrete realization scheme of several key steps of DGTD method is given and verified.Three kinds of numerical fluxes(central flux,upwind flux and partially penalized flux)and their applications in DGTD are studied in depth.The numerical flux is applied to the boundary of the elements for the exchange of field quantities.The semi-discrete DG matrix equations are obtained.And the integral expressions of the matrix coefficient are given.The iteration formulas of DGTD for general media are presented.Based on the equivalent principle and the Huygens principle,the plane wave is introduced at the boundary of the total-scattering field;and the equivalent line current and the magnetic current are calculated at the extrapolation boundary.And then the near field is extrapolated in far space.(2)The absorb boundaries of DGTD are studied systematically,including the first order Silver-Muller Absorb Boundary Condition(SM-ABC),Uniaxial anisotropic Perfectly Matched Layer(UPML),conformal UPML boundary.Firstly,the first-level and multi-level iterative formula of SM-ABC in DGTD are given.The truncation of SM-ABC in DGTD is realized.Start from the wave equation of UPML in frequency domain,a kind of truncation scheme of UPML layer with auxiliary variables B and D in DGTD is proposed first time.And the time domain iterative formulas are derived.In addition,a kind of wave equation and time-domain stepping formula of UPML with auxiliary variable P in DGTD are provided.Simulation results shown that both of these two kinds of UMPL techniques can implement the perfect absorption of outward traveling waves.Then from the DGTD equation in the orthogonal curve coordinate system,combined with the parameters conversion theory in different coordinate systems,the realization scheme of cylindrical and elliptical cylindrical UPML in DGTD and the specific time domain iteration formula are given.The axial thickness elliptical cylindrical UPML scheme which can be implemented easily in engineering is proposed and the realization process is given first time.Calculations prove that these three kinds of conformal absorption layers have good absorption effects for the outward traveling wave.Specific form of UPML can be selected according to the target shape to reduce the unknown and improve the calculation efficiency.(3)Tectonic process of the high order standard basis function and the high order hierarchical basis function in triangular element is given,and DGTD calculation based on higher order hierarchical basis function in two dimensional case is realized.The linear,the quadratic and the cubic type basis function in triangular element are given on the basis of Lagrangian function definition.Higher order hierarchical basis function in one dimensional case is used as the underlying linear basis function.On this basis,the higher level basis function is constructed in triangular element.The expressions of element matrices consist of basis function from the first order to the third order are presented.And the result of the integration of the element matrices is also calculated.The empirical formulas of calculating the stability of different order functions in DGTD are given.The influence of different order basis function on memory and calculating accuracy is discussed.(4)The Shift Operator Discontinuous Galerkin Time Domain(SO-DGTD)technique is proposed to deal with the electromagnetic properties of three kinds of common dispersive media.And this technique is applied to the analysis of radio transmission in dusty plasma.Based on the Maxwell equations with the electric displacement,the matrix equation is obtained by the integration by weighted for residual and the expansion by basis function.Differential discretizing matrix equation on time axis,the DGTD recursive formulas are obtained.On the other hand,taking into account that the common dispersive media model(Debye model,Drew model,Lorentz model)can be written as rational fractional form by the independent variable j?,applied the convert relationship from time domain to frequency domain(??t?j?),the constitutive relation is obtained.And then the time domain shift operator_tz is used instead of the time differential operator to deal with the time domain constitutive relation,and the discrete time domain iteration formula of the dispersive media is derived.Combined with the above equations,the calculation of the dispersive media SO-DGTD is implemented.Taking electromagnetic problem of weak ionized dusty plasma as an example,based on the modeling of the multi-physics commercial software COMSOL,The propagation of electromagnetic waves in the sheath was calculated by SO-DGTD method.The influence of dust concentration and dust particle radius on the radio wave transmission is analyzed,and the transmission characteristics of the sheath in the conditions of fluctuating flight velocity and altitude were also compared.