Introduction to the FDTD method

Transcription

1 Introduction to the FDTD method Ilkka Laakso Department of Electrical Engineering and Automa8on Tfy

2 Contents Principle of FDTD Deriva8on Basic proper8es Stability Dispersion Boundary condi8ons Advantages and weaknesses Examples

3 FDTD (= finite-difference time-domain) Principle 1. Start from Maxwell s equa8ons 2. Replace all deriva8ves with finite- difference approxima8ons 3. Done

4 Maxwell s curl equations

5 Central difference approximations Nota8on

6 Yee cell (Yee, 1966)

7 Derivation of FDTD update equations k + 1 µ H x k j j + 1

8 Derivation of FDTD update equations k + 1 µ H x k j j + 1

9 FDTD update equations k + 1 µ H x k j j + 1

10 H y E y FDTD update equations E x H z E x H z E y H y

11 FDTD update equations

12 Derivation of FDTD update equations from integral form of Maxwell s equations ( FIT method ) Use the mid- ordinate numerical integra8on method

13 Example: 1D FDTD Example1.m

14 Implementation in MATLAB % Magnetic field update equation Hy(1:K-1) = Hy(1:K-1) + Db(1:K-1).* ( Ez(2:K) - Ez(1:K-1) ); % Electric field update equation Ez(2:K-1) = Ez(2:K-1) + Cb(2:K-1).* ( Hy(2:K-1) - Hy(1:K-2) ); X coordinate H field indexing E field indexing

15 Contents Principle of FDTD Deriva8on Basic proper8es Stability Dispersion Boundary condi8ons Advantages and weaknesses Examples

16 Stability Courant- Friedrichs- Lewy (CFL) condi8on: In 3D: Numerical domain of dependence must include analy8cal domain of dependence v max Δt Δx

17 Numerical dispersion Accurate formula FDTD Numerical phase velocity v p = ω k

18 Numerical dispersion One- tenth of wavelength rule Maximum 8me step in 1D Maximum 8me step in 2D Maximum 8me step in 3D

19 Spectra

20 Anisotropic dispersion in 2D and 3D

21 Example: 2D FDTD 2D FDTD and anisotropic dispersion example3a.m Be careful with dielectric materials wavelength is shorter => finer cell size is needed example3b.m

22 Metal Absorbing boundary conditions In the basic form, FDTD can only model boxes, with ideally conduc8ng walls How to terminate the computa8on domain to model free space? FDTD is very good at modelling different materials: Berenger 1994: Make the walls of the box from an unphysical material that absorbs anything! Metal Metal Metal Absorbing material Metal ABCs are essen8al for any FDTD code Example4.m, example5a, 5b, 5c Metal Absorbing material Metal Metal

23 Contents Principle of FDTD Deriva8on Basic proper8es Stability Dispersion Boundary condi8ons Advantages and weaknesses Applica8ons

24 Advantages of FDTD (1) Simple equa8ons Can be parallelized easily Scales linearly with number of unknowns No need to solve equa8on systems => good for very large problems

25 ~ FDTD number of cells Absorbing boundary condi8ons in 3D 800 cells Yee 1966

26 Advantages of FDTD (2) Any kind of 8me- domain sources Geometry and boundary condi8ons are taken into account automa8cally. Any shape can be modeled easily Different media can be modelled naturally: non- linearity, inhomogeneity, anisotropy, complex geometry (metamaterials) Examples 6a, 6b FDTD Valuable (?) data

27 Weaknesses of FDTD (1) Not good for slow phenomena (huge number of 8me steps needed) Example: 1 mm grid resolu8on - > 8me step = 1.9e- 12 s Phenomenon las8ng 1 ms - > Number of 8me steps = 5e8 Curved shapes are problema8c (staircase approxima8on) High permikvity medium requires a fine grid One- tenth of wavelength rule

28 Comparison with analytic solution Radial component of the electric field in a sphere FDTD Analy8c solu8on

29 Weaknesses of FDTD (2) Computa8on domain must be finite Absorbing boundary condi8ons En8re computa8onal domain needs to be gridded (also empty space) Results depend on the choice of coordinate axes Error control Point of interest Empty space FDTD

30 Error in FDTD Trunca8on error from difference approxima8on (8me and space) Dispersion error, numerical anisotropy Unphysical Poyn8ng theorem, conserva8on of energy Floa8ng point (round- off) error Staircase approxima8on error Absorbing boundary condi8on error Modelling dielectric/lossy materials etc.

31 Contents Principle of FDTD Deriva8on Basic proper8es Stability Dispersion Boundary condi8ons Advantages and weaknesses Applica8ons

32 Applications of FDTD Radar cross sec8on and scamering Metamaterials Antenna analysis example 7a Electronic component design Electromagne8c compa8bility (EMC) Waveguides, resonators, filters example 7b Human exposure to EM waves

33 Human exposure to EM waves

34 Energy absorption 70 MHz 4.3 m 300 MHz 1 m 900 MHz 33 cm 2450 MHz 12 cm

35 Temperature rise Absorbed power = 0.4 W/kg Hirata, Laakso, et al 2013, Phys Med Biol

36 Power absorption versus temperature rise Hirata, Laakso, et al 2013, Phys Med Biol

37 When to use FDTD? Use FDTD first Use FDTD for making anima8ons Use FDTD for large heterogeneous geometries Use FDTD to model many things simultaneously Don t use FDTD at low frequencies Don t use FDTD with too large cell size Don t use FDTD if you need 99.9% accuracy