Perfectly Matched Layer Boundary Condition for Maxwell System ( - PowerPoint PPT Presentation

Perfectly Matched Layer Boundary Condition for Maxwell System ( using Finite Volume Time Domain Method ) 2005 Applied Math Seminar June, 14 th 2005 Krishnaswamy Sankaran

FVTD Method Berenger's PML Implementation Conclusion Outline of the talk  Introduction to Maxwell system & FVTD  Berenger's PML for Maxwell System  Implementation issues  Remarks & Conclusion 2

FVTD Method FVTD Method Berenger's PML Implementation Conclusion Maxwell System  Maxwell system describes solution to two divergence and two curl equations of electric (E) and magnetic (H) field.  In general for time domain analysis we concentrate on two maxwell curl equations describing space – time variation of these fields. H −  ∂  E ∇ X  −    E = J ∂ t E   ∂  H ∇ X   = K ∂ t 3

FVTD Method FVTD Method Berenger's PML Implementation Conclusion Maxwell System (continued...)  For our analysis we consider only homogeneous form of Maxwell curl equations .  = 0  J = 0  = K 0 H −  ∂  E ∇ X  = 0 ∂ t E   ∂  H ∇ X  = 0 ∂ t 4

FVTD Method FVTD Method Berenger's PML Implementation Conclusion Maxwell System (continued...)  Field quantities E and H are vector-valued functions 3 ℝ on space – time plane.  Spatial domain is (possibly unbounded.) 3  ⊂ ℝ  We consider finite time interval . = 0, T  ⊂ ℝ   Constitutive parameters: ε and μ are assumed to constant all over the domain. 5

FVTD Method FVTD Method Berenger's PML Implementation Conclusion Initial – Boundary Value problem  The initial – boundary value problem we are interested in here is to find the functions E and H for given t ∈  lim t  0  E  x ,t  = lim t  0  that . H  x ,t  = 0 ∀ x ∈  Initial – boundary Conditions Initial – boundary Maxwell system (perfect metallic, + value problem perfect magnetic, PML etc)  Above problem can be solved on computer taking into consideration of limited memory and time for processing. 6

FVTD Method FVTD Method Berenger's PML Implementation Conclusion Introduction to FVTD Method  FVTD stands for Finite Volume Time Domain  Conceived from Computational Fluid Dynamics (CFD), FVTD works on conservation laws for any hyperbolic system.  Basic idea is conservation of field quantities. 7

FVTD Method FVTD Method Berenger's PML Implementation Conclusion Finite Volume – Conservation Principle Flux of field into Flux of field out of i the cell [a,b] the cell [a,b] x=a x=b  The time rate of change of the total field inside the section [a,b] changes only due to the flux of fields into and out of the pipe at the ends x=a and x=b. 8

FVTD Method FVTD Method Berenger's PML Implementation Conclusion Maxwell system in Conservative Form Q t  F 0  Q  x  G 0  Q  y = 0 T T =  H x , H y , E z  TM case Q =  Q 1, Q 2, Q 3  T − E x , − E y , H z  TE case T F 0  Q   0, − Q 3, − Q 2  = T G 0  Q   Q 3, 0, Q 1  = For our analysis we use only TM case T F 0  Q   0, − E z , − H y  = T G 0  Q  =  E z , 0, H x  9

FVTD Method FVTD Method Berenger's PML Implementation Conclusion Finite Volumes in 3D Face 2 Face 3 Face 1 Bary-centre (BC) Face-centre (FC) Node Face 4 10

FVTD Method FVTD Method Berenger's PML Implementation Conclusion Finite Volumes in 2D Neighbour 3 Neighbour 2 Bary-centre (BC) Face-centre (FC) Neighbour 1 Node 11

FVTD Method FVTD Method Berenger's PML Implementation Conclusion Edge Fluxes Godunov 1 st Order q R q i+1 q L q i Outgoing flux MUSCL 2 nd Order q R Incoming flux q L q i+1 q i 12

FVTD Method FVTD Method Berenger's PML Implementation Conclusion Flux approximation Piecewise constant Piecewise linear flux approximation flux approximation 13

FVTD Method Berenger's PML Berenger's PML Implementation Conclusion Berenger PML  The method used in Berenger PML to absorb outgoing waves consists of limiting computational domain with an artificial boundary layer specially designed to absorb reflectionless the electromagnetic waves. Γ ∞ Ω 4 Ω 3 Ω 2 PML Free space Object Ω 5 Ω 1 Γ b Incident wave Scattered wave Ω 6 Ω 7 Ω 8 14

FVTD Method Berenger's PML Berenger's PML Implementation Conclusion Berenger PML  The computational domain is divided into two parts. Free space or vacuum – classical Maxwell equations. Absorbing Layer – modified Maxwell equations. Modified Maxwell equation  ∂  H  ∇ X  E   H  H = 0 ∂ t  ∂  E − ∇ X  H   E  = E 0 ∂ t  σ H and σ E are magnetic and electric conductivities respectively. 15

FVTD Method Berenger's PML Berenger's PML Implementation Conclusion Modified Maxwell system  Modified Maxwell system can be considered as classical Maxwell system with source terms. To analyse the modified eqns at continuous levels leads to the condition: σ H = σ E = σ . Modified Maxwell equation  ∂  H  ∇ X  E    = H 0 ∂ t In FVTD formulation  ∂  E these terms are − ∇ X  H    E = 0 considered as ∂ t source terms σ H = σ E = σ enables reflectionless transmission of a plane wave propagating normally across the interface between free space and outer boundary. 16

FVTD Method Berenger's PML Berenger's PML Implementation Conclusion Berenger's PML  J. P. Berenger published (J. Comp. Physics No. 114 – year 1994) this novel technique called PML in 2D case.  With this new formulation, the theoretical reflection factor of a plane wave striking a vacuum – layer interface is zero at any incidence angle and at any frequency.  We model this PML in 2D set-up . We make use of 2D Maxwell equations with TM formulation. Generalising to 3D full wave analysis is straightforward. 17

FVTD Method Berenger's PML Berenger's PML Implementation Conclusion Berenger split field formulation  We split E z field into two subparts: E zx and E zy . Hence we have four equations in modified Maxwell equations.  ∂ H x  ∂ E zx  E zy    y H x = 0 ∂ t ∂ y  ∂ H y − ∂ E zx  E zy    x H y = 0 ∂ t ∂ x  ∂ E zx − ∂ H y   x E zx = 0 ∂ t ∂ x  ∂ E zy  ∂ H x   y E zy = 0 ∂ t ∂ y  Magnetic and electric conductivities are also split into σ Hx , σ Hy , σ Ex and σ Ey with conditions σ Hx = σ Ex = σ x and σ Hy = σ Ey = σ y . 18

FVTD Method Berenger's PML Berenger's PML Implementation Conclusion σ x and σ y – Physical Interpretation  Choice of σ x and σ y is very critical to obtain perfectly transparent vacuum - layer interfaces for outgoing waves.  σ x can be interpreted as absorption coefficient along x- direction. Correspondingly σ y is along y- direction. If  e x isthe normal direction for theinterface between free space − PMLmediumthen  = 0 ∀ i and ∀ if  y = 0  = reflectioncoefficient  i = incidence angle  = wave frequency Similarly if  e y isthe normal direction for theinterface between free space − PML mediumthen  = 0 ∀ i and ∀ if  x = 0 19

FVTD Method Berenger's PML Berenger's PML Implementation Conclusion Conductivity choices  Computational domain is bounded in all sides by artificial absorbing layers namely Ω 1 to Ω 8 .  1 =  x , y  ; y ∈ [− b ,b ] , x ∈ [ a , A ]  =  1 ∪ ... ∪ 8 where  2 =  x , y  ; y ∈ [ b , B ] , x ∈ [ a , A ]  3 =  x , y  ; y ∈ [ b , B ] , x ∈ [− a ,a ] Γ ∞ Ω 4 Ω 3 Ω 2 PML Free space Object Ω 5 Ω 1 Γ b Incident wave Scattered wave Ω 6 Ω 7 Ω 8  Also to avoid parasitic reflections on the interface of the free space and PML medium, we take σ y = 0 in Ω 1 and σ x = 0 in Ω 3 etc. 20

FVTD Method Berenger's PML Berenger's PML Implementation Conclusion Conductivity choices (continued...)  Based on the discussions before we can more precisely define conductivity choices in different portions of artificial boundary.   =  x  e x   y  e y  0  A − a  n x − a  1  = e x   0  B − b  n y − b  3  = e y   =  1 in  1    =  3 in  3     =  1     3 in  2  Choice of σ 0 and n play a vital role in formulating reflectionless boundary condition. Different possibilities are disscussed here. 21

FVTD Method Berenger's PML Berenger's PML Implementation Conclusion Conductivity choices (continued...)  One another possible choice of σ 0 can be done as presented paper of F. Collino, P.B. Monk (Comput. Methods Appl. Mech. Engrg. No. 164 year 1998 pg 157 – 171.) 2  c  =  layer length = 1 wavelength    0    2 x − a  x   = ∀ x  a  parabolic − law  e x ,   0    2 y − a  y  = ∀ y  b  e y ,  3 − 2 , 10 − 3 , 10 2  log e  R 0 − 1  − 4  0  = R 0 = 10 22