THE FINITE-DIFFERENCE TIME-DOMAIN (FDTD) METHOD PART IV

Transcription

1 Numerical Techniques in Electromagnetics ECE 757 THE FINITE-DIFFERENCE TIME-DOMAIN (FDTD) METHOD PART IV The Perfectly Matched Layer (PML) Absorbing Boundary Condition Nikolova

2 1. The need for good absorbers good performance of the absorbers is crucial for (1) the accuracy of frequency-domain responses (2) reducing the size of the computational domain (3) the analysis of low-rcs targets, low-reflection coatings, matched loads, etc. numerical errors below -40 db (1/100) always desirable, sometimes -80 db Mur and Liao absorbers provide effective reflection coefficients of about 0.5 % to 5 %: errors above -40 db are common Berenger publishes his first work on PML in 1994 reporting reflections of about 3000 times less than Mur s 2 nd order ABC! Nikolova

3 2. Theory of plane wave diffraction: review We know that for reflection-free propagation through the interface between two mediums, their intrinsic impedances must be matched. The intrinsic impedance of a fictitious lossy medium which has both electric and magnetic conductivity is μ μ jμ σ e σ 1 jσ m m μ ηl If ε, μ ωμ ηl ε ε jε ω ω ε 1 jσ e ωε Let the lossy region be region 2 onto which plane waves are incident from region 1. Region 1 is loss-free and with real constitutive parameters ε, μ. Its intrinsic impedance is then η μ/ ε. Let ε ε, μ μ. The propagation constants are σ e σ m γ jω εμ and γl jω εμ 1 j 1 j ωε ωμ Nikolova

4 2. Theory of plane wave diffraction, cont. If the condition σ e ε σ m μ! impedance matching condition is observed in the lossy medium, then ηl η, and a plane wave normally incident upon the interface is not reflected back! Moreover, the velocity of propagation is the same as in region 1: γ l jω με + ησe and the medium is dispersion-free despite its losses. β α Nikolova

5 2. Theory of plane wave diffraction, cont. At oblique incidence, it is not enough to ensure that the impedance matching condition is observed. For eample, recollect that the reflection coefficients for perpendicular and parallel polarization of the wave are Γ E η2cosθi η1cosθt E η cosθ + η cosθ r 0 i 0 2 i 1 t The angles of incidence and transmission Θ i and Θ t are related through the phase matching condition: γ1sinθi γ1sinθr γ2sinθt which ensures the continuity of the tangential to the interface field components. When η 1 η 2, reflection is zero only if the angles of incidence and transmission are the same! We net see how all these conditions are observed in the PML medium. Nikolova Γ η2cosθt η1cosθi η cosθ + η cosθ 2 t 1 i

6 3. Berenger s Perfectly Matched Medium: TE Case Mawell s equations for the TE z case (source-free): H z E y E μ + σmh z t y E H z ε + σee t y Ey H z ε + σeey t Berenger splits the H z field component: H z Hz + Hzy so that (look at the 1st equation), the -derivative of E generates H z, and the y-derivative of E generates H zy. He also introduces different specific conductivities to accompany the split terms. Nikolova

7 3. Berenger s Perfectly Matched Medium: TE Case, cont. H μ t H μ t E ε t z zy H E + σm z + σ + σ E ey my H y ( H + H ) z y Ey ( Hz + Hzy) ε + σeey t zy E y We net study the plane wave propagation in Berenger s medium. zy Nikolova

8 4. Plane Waves in Berenger s Medium: TE Case Let a time-harmonic plane TE z wave propagate as shown in the figure at an angle Φ with respect to the -ais. The E-field then forms an angle Φ with respect to the y-ais. y 1 1 j ( t v v y) y E E0sinφ e ω 1 1 j ( t v v y) y Ey E0 cosφ e ω z z y j ( t v v y) H H e ω y j ( t v v y) H H e ω zy zy E φ z H zˆ H z φ P The constants v and v y are comple. They describe the wave behavior in space and can be viewed as comple velocities. We find them by substituting the above field components in Berenger s TE z equations. Nikolova

9 4. Plane Waves in Berenger s Medium: TE Case, cont. E ( H z + H ) zy ε + σeye t y Ey ( Hz + Hzy) ε + σeey t H E z y μ + σmh z t H zy E μ + σmyh zy t y σ ey 1 ε j E0 sin φ vy ( Hz 0 + Hzy 0) ω σ e 1 ε j E0 cos φ v ( Hz 0 + Hzy 0) ω σ m 1 μ j Hz0 v E 0 cosφ ω σ my μ j Hzy0 vy E sinφ ω 1 0 We epress H z0 and H zy from the last two equations and substitute them in the 1 st 0 two. Nikolova

10 4. Plane Waves in Berenger s Medium: TE Case, cont. We obtain two equations for the constants v and v y : 1 με σ ey v cosφ v sinφ 1 j sinφ 1 vy ωε + σ m σ my 1 j 1 j ωμ ωμ 1 1 με σ e v cos vy sin 1 j φ φ cosφ 1 + v ωε σ m σ my 1 j 1 j ωμ ωμ This system gives two solution sets: we choose the one with 1 and v 1 y being with positive real part, so that the wave propagates along the positive and y aes. 1 y v Nikolova

11 4. Plane Waves in Berenger s Medium: TE Case, cont e v σ j cos φ vg ωε 1 1 ey v σ y 1 j sin φ vg ωε! where v w 1, με σ e 1 j ωε, σ m 1 j ωμ G w 2 φ + wy 2 w cos sin φ, y 1 1 σ ey j ωε. σ my j ωμ We can now return to the system in slide 9, substitute v and v y, and obtain and. H z0 H zy 0 Nikolova

12 4. Plane Waves in Berenger s Medium: TE Case, cont. H z0 E 0 w cos ηg 2 φ H zy0 E 0 w y sin ηg 2 φ η μ ε + G Hz0 Hz 0 Hzy 0 E0 η Thus, the intrinsic impedance of the wave in Berenger s PML medium is E η G η 0 PML H! z0 Nikolova

13 4. Plane Waves in Berenger s Medium: TE Case, cont. Each of the wave components is of the form Re-arranging: 1 v 1 σ e ψ ψ0 ep( jωt) ep jω 1 j cosφ vg ωε 1 σ ey ep jω 1 j sin φ y. vg ωε 1 v y cosφ + ysinφ η ψ ψ0 ep jω t ep σe cosφ vg G η ep σ ey sin φ y G Nikolova

14 5. Impedance Match at the Interface with PML If the conditions σ e σ m σ ey σ my, ε μ ε μ! are fulfilled, then 2 2 w w 1 G wcos φ+ wysin φ 1 y η η PML The last equation shows that the impedance of the PML medium is equal to that of the loss-free medium regardless of the angle of propagation: impedance match is achieved for plane waves of any angle of incidence. Nikolova

15 5. Impedance Match at the Interface with PML, cont. The wave in the PML propagates as cosφ + ysinφ ψ ψ0 ep jω t ep η( σe cosφ + σey sinφ y) v E φ z H zˆ H z phase delay: jω( t rv / v ) j( ωt kr ) y v v v( ˆcosφ + yˆsin φ), k φ P 2 ω v v v retardation time is r r( vr ˆ ˆ) τ vfront v r vˆ r v 2 v v attenuation Nikolova z y v vfront v/( vr ˆ ˆ) y r φ v v

16 5. Impedance Match at the Interface with PML, cont. We now have to ensure that the continuity of the tangential field components is achieved by matching their phase terms along the ais tangential to the boundary. Assume that the boundary is along the y-ais (unit normal is ). Then, the matching of the phase terms at the interface along y requires (see slide 13 or 15) ep( sin ) σ ey jω με φ y ep jω με 1 j sinφ y ωε at 0 This can be achieved only if σ ey 0, which in accordance with the impedance-match condition means also that σ my 0. There will be no attenuation along the tangential y-ais. On the other hand, we require maimum attenuation along the -ais. We choose appropriate functions for σ e ( ) and σ m ( ) which satisfy the impedance-match condition. Nikolova

17 6. Berenger 2-D PML: TE z Case (1) (1) (2) (2) e m ey my PML( σ, σ ; σ, σ ) (2) (2) ey σ my PML(0,0; σ, ) (2) (2) (2) (2) e m ey my PML( σ, σ ; σ, σ ) y PML( σ e, σ m ;0,0) (1) (1) PML( σ e, σ m ;0,0) (2) (2) z (1) (1) (1) (1) e m ey my PML( σ, σ ; σ, σ ) (1) (1) ey σ my PML(0,0; σ, ) (2) (2) (1) (1) e m ey my PML( σ, σ ; σ, σ ) Nikolova

18 6. Berenger 2-D PML: TE z Case, cont. When a PML interface is orthogonal to the ais (its unit normal is along ), the wave components must attenuate along. This is accomplished by introducing σ e and σ m. To ensure the continuity of the tangential field components, σ ey and σ my must be zero. On the contrary, for an interface of unit vector along y, nonzero σ ey and σ my are introduced, while σ e and σ m are zero. At corner regions, all four loss parameters are nonzero. Nikolova

19 7. Berenger 2-D PML: TM z Case The analysis for the TE case can be repeated for a TM z wave, and it follows along the same lines. The results are dual. We give a summary below. The split equations for the TM z case are E H z y H ( Ez + E ) zy ε + σeez μ + σmyh t t y Ezy H ε + σ H y ( Ez + Ezy) eyezy t y μ + σmh y t The PML matching conditions are the same and the 2-D PML regions are constructed as in slide 17. Nikolova

20 8. Berenger s 3-D PML In 3-D, all si field components are split according to the field component derivatives generating them. The procedure of splitting is identical to the 2-D cases. ε + σ ey Ey ( Hz + H zy ) μ σ + my H y t y t y E + E ε + σ ez Ez ( Hy + H yz ) μ σ + mz Hz t z t z Ey + Eyz ε + σ e Ey ( Hz + H zy) μ σ + m Hy t t Ez + Ezy ε + σ ez Eyz ( Hy + H z) μ σ + mz H z t z t z E + E ε + σ e Ez ( Hy + H yz) μ σ + m H t t E + E ε + σ ey Ezy ( Hy + H z) μ σ + my Hzy Ey + Ez Nikolova t 2009 y t y 20 ( ) z zy ( ) ( ) ( ) y y z ( ) z y yz ( )

21 8. Berenger s 3-D PML, cont. The matching conditions at a planar interface between the lossfree computational region and the PML require that the specific conductivities along the unit normal of the interface must be nonzero and satisfying the impedance-match condition σ en ε σ mn μ where n denotes the ais orthogonal to the planar interface. The other two pairs of conductivities (along the aes which are tangential to the interface) are set equal to zero. In a dihedral corner where two orthogonal PMLs overlap, two pairs of conductivities are nonzero the ones which are nonzero in the neighboring PMLs. In a trihedral corner where three PMLs overlap, all si conductivities must be nonzero. Nikolova

22 8. Berenger s 3-D PML, cont. σ ey, σ my 0 σ ez, σ mz 0 σ e, σ m 0 σ ey, σ my, σ ez, σ mz 0 σ e σ m 0 z y σ ey, σ my 0 σ e σ m σ ez σ mz 0 Nikolova

23 8. Berenger s 3-D PML, cont. Discrete form of the PML equations (eample for the E y, H y ): nˆ yˆ ε σ E 1 ξe, j E Δt / ε + ey Ey ( Hz + H zy ) ke, j, kh, j, t y 1+ ξe, j 1+ ξe, j n+ 0.5 n+ 0.5 H 1 z H i, j, k z n+ E n E i, j 1, k Ey k i, j, k E, j Ey + k i, j, k H, j Δy μ σ + my H E + t y ( E ) y z zy n n E z E i, j 1, k z n+ H n H + i, j, k Hy k i, j, k H, jhy k i, j, k E, j Δy k ξ e, j σ ey, jδt 2ε y jδy 1 ξ Δt / μ,, H m, j H H, j ke, j 1+ ξm, j 1+ ξm, j ξ m, j σ Δt my, j 2μ y ( j+ 1/2) Δy Nikolova

24 8. Berenger s 3-D PML, cont. Discrete form of the PML equations as first proposed by Berenger, eponential time stepping (eample for the E y, H y ): ε t + σ ey E H + y ( H ) y z zy E E σ, / 1 ey jδt ε E ke, j E, j, H, j σ eyδy k e k ( ) E k E + k H H n+ 1 E n E n+ 0.5 n+ 0.5 yi, j, k E, j yi, j, k H, j zi, j, k zi, j 1, k μ t + σ my H E + y ( E ) y z zy H H σ, / 1 my jδt μ H kh, j H, j, E, j σ myδy k e k ( ) H k H k E E n+ 0.5 H n 0.5 H n n yi, j, k H, j yi, j, k E, j zi, j+ 1, k zi, j, k Nikolova

25 9. PML Loss Parameters Theoretical reflection from the PML The PML is usually backed by a PEC wall. The reflected signal undergoes reflection at the PML termination but also undergoes substantial attenuation corresponding to double the thickness of the PML d. In a PML layer where constant attenuation is assumed along the normal direction only (the tangential conductivities are zero), the reflection coefficient becomes R( φ) ep 2σ ηcosφ d ( en ) φ ˆn d Reminder (see slide 15): If n, then ψ ( y, ) ψ 0 ep( σ eη cos φ ) ep jω t cosφ + ysinφ v Nikolova

26 9. PML Loss Parameters, cont. R(Φ) is the PML reflection error. It gives the relative magnitude of the spurious reflected wave, which enters back into the computational domain. The larger d and σ en are, the less the reflection. However, the angle of incidence Φ plays an important role, too. When Φ 90 deg., R 1! At grazing angles of incidence, the PML is ineffective at the corner regions of the computational domain. In practice, the Berenger PML is placed sufficiently far from sources and guiding structures so that the plane-wave components of the field impinge upon the interface at angles smaller than 90 deg. Nikolova

27 9. PML Loss Parameters, cont. PML in Discrete Space Theoretically, reflectionless wave transmission should take place through the PML interface, regardless of the local step discontinuity in the normal conductivities σ en and σ mn. In practice, however, spurious numerical reflections do arise, because of the finite spatial sampling of the field. Therefore, we can not set σ en and σ mn to be large constant numbers throughout the PML. The conductivities are made functions of the PML depth: they have to be very small close to the PML interface (in order to ensure as little as possible spurious reflection), and then increase as quickly as possible toward the PEC termination wall (in order to ensure sufficient attenuation). Nikolova

28 9. PML Loss Parameters, cont. Assume that is the position measured from the PML interface inward toward its PEC termination. Then, for σ e ( ) d R( φ) ep 2ηcos φ σen( d ) 0 There are various profiles for the conductivity. (a) Polynomial grading σ e d m σ ema σ (0) 0, σ ( d) σ e e ema The bigger m is, the smoother the change of σ e close to the interface. But, then, the steeper its slope is close to the PEC walls: spurious numerical reflections occur deeper in the PML. Nikolova

29 9. PML Loss Parameters, cont. We then have to bring down σ e,ma. This, however, may lead to insufficient attenuation. Alternatively, we can keep σ e,ma large but increase the PML depth d to allow for acceptable slopes at all points deep in the PML. This, however, means increase of the required computational resources. Designing an efficient PML is not an easy task! The reflection coefficient with polynomial grading is [ ema ] R( φ) ep 2ησ dcos φ /( m+ 1) Typical optimal values: 2 m R(0) 10 (for d 10 Δ ), 10 (for d 5 Δ) Nikolova

30 9. PML Loss Parameters, cont. When R(0), m, and d are set, we can compute σ e,ma : σ ema [ R ] ( m+ 1)ln (0) 2η d (a) Geometric grading The PML loss factor is defined as σ e ( ) ( 1/ Δ g ) σ 0 d/ Δ σ e (0) σ0, σe ( d) σ0g scaling factor conductivity at interface ( d/ Δ ) R( φ) ep 2ησ 0Δ g 1 cos φ/lng Nikolova

31 9. PML Loss Parameters, cont. σ 0 must be small for less spurious reflection from the interface. The scaling g > 1 determines the rate of increase of the conductivity. Large g s flatten the conductivity profile near the interface and make it steeper deeper into the PML. Usually, 2 g 3 If R(0), g and d are given, we can compute σ 0 : σ [ R ] ln (0) ln g 2ηΔ g 1 0 d/ ( Δ ) Nikolova

32 9. PML Loss Parameters, cont. There is another implementational detail concerning the computation of the conductivity at a mesh point: it is given by the average value in the cell around the inde (L) location: σ en ( L+ 0.5) Δ 1 ( L) σ en( ) d Δ ( L 0.5) Δ Thus, for a polynomial grading of order m in a PML, which is N-cell thick, σ [ R ] σ ln (0) ( m+ 1)2 N 2 ηδn ( mn, ) ema e (0) m+ 1 m m+ 2 m σ σ ( mn, ) ( mn, ) m 1 m 1 e ( L> 0) e (0) (2L+ 1) (2L 1) Nikolova

33 9. PML Loss Parameters, cont. For the geometric grading of scaling g in a PML of N cells, σ ( gn, ) e (0) σ e0 N [ R ] g 1 (1 g) ln (0) ln g 2 ηδg ( 1) ( gn, ) ( gn, ) L1/2 e ( L> 0) e (0) g σ σ Nikolova

34 Important topics not mentioned in this course FDTD numerical dispersion errors FDTD on curvilinear grids, conformal FDTD (C-FDTD) schemes FDTD in dispersive and anisotropic media FDTD in nonlinear and gain materials Integrating lumped elements with the FDTD full-wave analysis Ecitation schemes for enhanced convergence Near-to-Far-Field transformation for antenna radiation patterns Modified implicit FDTD schemes the FDTD-ADI Eigen-mode analysis of waveguides S-parameter analysis with FDTD Nikolova