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
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