計(jì)算電磁學(xué)簡(jiǎn)介林志立

1、Dr. Zhili LinA Brief Introduction to Computational Electromagnetics Beihang University 2009-11-02Email: Basic Theory in ElectromagneticsMaxwell Equations (1873, James Clerk Maxwell)Amperes Law with Maxwells correctionFaradays law of inductionGausss lawGausss law for magnetismConstitutive relationsJa

2、mes Clerk Maxwell (18311879)Equation of continuity Mathematical methods for solving electromagnetic problems1.Analytical Methods separations of variables; series expansion; conformal mapping; perturbation methods; transform methods, etc.2.Numerical Methods finite difference method (FDM, including FD

3、TD); finite element method (FEM); method of moments (MoM); transmission-line modeling; Monte Carlo method, etc.Why choosing Numerical Methods ?lThe ordinary differential equations that have close-formed analytical solution are very limited and only for a few specific cases. lAlthough explicit soluti

4、ons can offer a lot of benefits, they are not easily obtained if the problem is complicated in geometrical structure.lWhen the applied medium is complex (inhomogeneous, dispersive), it is often hardly possible to solve the equation analytically.The possibility of Using Numerical Methods - Thanks to

5、the Computer Tech.lIn 60s of 20th century, the modern computer is invented. Since then, obeying the Moores law, computers become more and more powerful.lThe CPU runs faster and faster, the memory becomes larger and lager, the size of a computer becomes smaller and smaller, while the cost of a comput

6、er becomes cheaper and cheaper. lBecause of this, people tend to find some methods that are simple in formula but require large amounts computation that can be easily handled in a computer. As the methods are too many to be all covered, in the following , I will only briefly introduce two methods: F

7、inite-Difference Time-Domain (FDTD) Method Finite Element Method (FEM) Finite-Difference Time-Domain MethodThe Standard Yees CellYee, K. S., “Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media,”IEEE Trans. Antennas Propagat., vol. 14, 1966, pp. 302-

8、307.2627 citations by 16.02.2007 Source: ISI Web of ScienceExt 10HyzFDTD-1D ProblemFor Ampere Law, 1-D Problem:To put the equation in a computer, take the finite-difference approximations of the partial derivatives in time and space:1/21/2,( )( )nnxxxEz tEkEktt,(1/2)(1/2)nnyyyHz tHkHkzxFDTD-Discreti

9、zation1/21/20( )( )(1/2)(1/2)nnxxnnyyEkEktHkHkx11/ 21/ 20(1/ 2)(1/ 2)(1)( )nnyynnxxHkHktEkEkxThis is a time-domain method. Each new value of the electric field E or the magnetic field H is determined by the previous valuesSimilarly,FDTD-Computer AlgorithmThe k represents the location in an array in

10、a computer while n represents timeCalculate En+1/2Calculate Hn+1n=n+1Each time step represents an increment in the total time T = n t.FDTD-Computer AlgorithmFDTD-Parameters and StabilitySelection of the parameters of simulation Cell Size Criterionmaxmaxmaxmin1, and 10 xyz0minmaxmaxmaxrrcf Courant St

11、ability:2221111zyxct1D simulation of an EM pulse propagation in free spaceclear;clf;KE=200;for i=1:KE ex(i)=0; hy(i)=0;endkc=5;t0=80;spread=24; NSTEPS=550; figure(1); plot(ex); hold on; plot(hy); axis(0 KE -1 1); hold off; pause(0.01); endT=0;for n=1:NSTEPS T=T+1; for k=2:KE ex(k)=ex(k)+0.5*(hy(k-1)

12、-hy(k); end pulse=exp(-0.5*(t0-T)/spread)2); ex(kc)=pulse; for k=1:KE-1 hy(k)=hy(k)+0.5*(ex(k)-ex(k+1); endFDTD Codes for InitializationMain Computing LoopVisualizationInserting the source1D simulation of an EM pulse propagation in free space1D simulation of an EM pulse propagation in free space1D s

13、imulation of an EM pulse propagation in free space1D simulation of an EM pulse propagation in free space1D simulation of an EM pulse propagation in free space1D simulation of an EM pulse propagation in free space1D simulation of an EM pulse propagation in free space1D simulation of an EM pulse propa

14、gation in free space1D simulation of an EM pulse propagation in free spaceApplicators can be simulated in FDTD by specifying the material and the source of energy. Here is a simple dipole antenna.A simple dipole antenna -3D simulationThe portion that is to be metal can be specified by just holding t

15、he E field to zero.The excitation comes from just specifying the E field in the gap.The FDTD method will determine the H field, which is an indication of the current in the dipole.3D simulation of a simple dipole antennaThe first set of slides shows the H fields next to the metal of the dipole arms,

16、 which are an indication of the current.The following set of slides shows the H fields next to the metal of the dipole arms, which are an indication of the current.The next set of slides shows the E field in the plane of the gap as it radiates away from the antennaUp until now, we have not discussed

17、 the boundary conditions at the edges of the problem space. These are necessary to keep unwanted reflections from coming back.FDTD -Absorbing Boundary Condition (ABC)Probably the best solution is the perfectlymatched layer (PML) which absorbsout-going waves.The reflection of an outgoing wave is dete

18、rminedby the reflection coefficient ABABFDTD-Absorbing Boundary Condition (ABC)We assume and are complex. It is the imaginary part that leads to absorption,*FmFmDmj0*FmFmHmj0ABC- Perfectly Matched Layer, ,mx y zThere are two conditions to form a PML:0m*Fx*Fx1. The impedance going from the background

19、 medium to the PML must be constant:2. The direction perpendicular to the boundary, the x direction for example, must be the inverse of the other directions:*Fx1*Fy*Fx1*FyABC- Perfectly Matched LayerThe following selection of Simple parameters satisfies these requirements:FmFm1Dm0Hm0D0ABC- Perfectly

20、 Matched Layer0m*Fx*Fx1(x) / j01(x) / j01An outgoing wave sees a constant impedance as its going into the PML. The conductivity causes it to be absorbed once its in the PML.The values of are gradually increased as they go into the PMLFt DH *DEFt HETo implement the PML, we will assume that there are

21、“fictitious” values of and that we can attach to the Maxwells equations.ABC- Perfectly Matched LayerPMLDispersive MaterialThe radiation from a point source is absorbed by the PML.Simulating Dispersive Media202200()( )2srj Example- Lorentz media:10120121012012ee( ) ,eejjrjja Zaa Zaaab Zbb ZbbbDE 1121

22、0110210()nnnnnnbbbaaaDDDEEEUpdating Equation in FDTDHow to ApproximateZ-transform00 ( )( )( )1rtt DHDEHESimulating Dispersive MediaVarious Approaches for Modeling Lorentz Media By FDTD012012,a a a b b bMSE approach :( )( )rrError Recommended References:lA. Taflove and S. C. Hagness, Computational El

23、ectrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. Norwood, MA: Artech House, 2000. (經(jīng)典大作）lD. M. Sullivan, “Electromagnetic Simulation Using the FDTD Method,” IEEE Press, 2002.（入門書籍）Software:Books:Opti-FDTD，加拿大Optiwave公司開發(fā)，時(shí)域光子學(xué)模擬仿真； Meep (or MEEP) ，MIT 開發(fā)，模擬電磁體系； 德國(guó)IMST公司的 EMPIRE，射頻（R

24、F）器件設(shè)計(jì) ；F2P ，KTH 副教授Min Qiu (浙大校友）開發(fā)，for photonic devices； XFDTD, 美國(guó) REMCOM 公司 ，高頻電磁分析模擬軟體,計(jì)算SAR值； CFDTD，全名Conformal FDTD，by Wenhua Yu, 復(fù)雜金屬邊界；Finite Element Method（FEM)Example: Electrostatic Problems1-D ProblemTwo Metal PlateBoundary conditions:Analytical solutions:FEM-Discretization of DomainEleme

25、nt Along -Axis (Master Element)Element Along x-Axis.Coordinate Transformation:Finite Element MethodFinite Element MethodWeighted-integral Equationw (x) - Weight functionFinite Element Method12 wNwNEquation for the e-th element :Finite Element MethodFinite Element MethodFinite Element MethodDivided i

26、nto 4 SegmentsFEM-2D ProblemTwo-Dimensional Boundary-Value Problems2-D ProblemFEM-Two-DimensionQuantity u:FEM-Two-DimensionwhereEquations to be solved:FEM-Two-DimensionFinite ElementsFinite element mesh using triangular elementsCoarse Elements FEM-COMSOL multiphysicsExample: Modeling Photonic Crysta

27、lGaAs RodAir Back GroundMenu Panel of the COMSOLMenus COMSOL - Modeling Photonic CrystalSelecting the Models COMSOL - Modeling Photonic CrystalSet Scalar ParametersSuch as wavelength. COMSOL - Modeling Photonic CrystalDRAW MODE- Drawing Domain BoundariesSet length and size COMSOL - Modeling Photonic CrystalPOINT MODE- Setting Properties of Points COMSOL - Modeling Photonic CrystalBOUNDARY MODE- Setting Properties of Boundaries COMSOL - Modeling Photonic CrystalSUBDOMAIN MODE- Setting Propertie