Perfectly Matched Layers for Maxwell Equations in Plasmas E. B´ ecache, P. Joly, M. Kachanovska POEMS, INRIA, ENSTA ParisTech December 3, 2014 E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 1 / 32
Outline Outline 1 Introduction Maxwell Equations in Cold Plasma Model Problems: Isotropic Dispersive Model and Uniaxial Plasma Model (L.Colas) Introduction into the PML 2 PMLs for Plasmas Drude Model for Metamaterials: General Dispersive Isotropic Model Uniaxial Cold Plasma Model in 2D Uniaxial Cold Plasma Model in 3D 3 Conclusions E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 2 / 32
Introduction Maxwell Equations in Cold Plasma Model Problem Maxwell Equations in the Frequency Domain in R 3 In the frequency domain in R 3 E + ω 2 − curl curl ˆ c 2 ǫ ( ω )ˆ B 0 = (0 , 0 , B 0 ) E = 0 . z y x E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 3 / 32
Introduction Maxwell Equations in Cold Plasma Model Problem Maxwell Equations in the Frequency Domain in R 3 In the frequency domain in R 3 E + ω 2 − curl curl ˆ c 2 ǫ ( ω )ˆ B 0 = (0 , 0 , B 0 ) E = 0 . Cold plasma dielectric tensor: z ω 2 ω 2 y p ω c p 1 − − i 0 x ω 2 − ω 2 ω ( ω 2 − ω 2 c ) c ω 2 ω 2 p ω c ǫ ( ω ) = p i 1 − 0 ω ( ω 2 − ω 2 ω 2 − ω 2 c ) c ω 2 p 0 0 1 − ω 2 Single-species cold plasma N e ( x , y , z ) concentration of particles e particle charge particle mass m e � N e e 2 ω p = plasma frequency m ǫ 0 ω c = e B 0 algebraic cyclotron frequency m e c E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 3 / 32
Introduction Maxwell Equations in Cold Plasma Two limit cases ω 2 ω 2 p ω c p 1 − − i 0 ω 2 − ω 2 ω ( ω 2 − ω 2 c ) c ω 2 ω 2 p ω c ǫ ( ω ) = p 1 − 0 i ω ( ω 2 − ω 2 ω 2 − ω 2 c ) c ω 2 p 0 0 1 − ω 2 ω c = 0 (i.e. B 0 = 0) 1 ω 2 p 1 − 0 0 � � ω 2 ω 2 ω 2 p ǫ ( ω ) = = 1 − I 3 p 0 1 − 0 ω 2 ω 2 ω 2 p 0 0 1 − ω 2 This is an isotropic dispersive case . E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 4 / 32
Introduction Maxwell Equations in Cold Plasma Two limit cases ω 2 ω 2 p ω c p 1 − − i 0 ω 2 − ω 2 ω ( ω 2 − ω 2 c ) c ω 2 ω 2 p ω c ǫ ( ω ) = p 1 − 0 i ω ( ω 2 − ω 2 ω 2 − ω 2 c ) c ω 2 p 0 0 1 − ω 2 ω c = 0 (i.e. B 0 = 0) 1 ω 2 p 1 − 0 0 � � ω 2 ω 2 ω 2 p ǫ ( ω ) = = 1 − I 3 p 0 1 − 0 ω 2 ω 2 ω 2 p 0 0 1 − ω 2 This is an isotropic dispersive case . ω c → ∞ 2 1 0 0 0 1 0 ǫ ( ω ) = ω 2 p 0 0 1 − ω 2 This is an anisotropic dispersive case (uniaxial plasma (L. Colas)) E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 4 / 32
Introduction Model Problems Two Problems Recast in the Time Domain Generalized hyperbolic system: Anisotropy � �� � Anisotropy+Dispersion d ���� � U ( x , t ) ∈ R m , A i , B ∈ R m × m , i = 1 , . . . , d . ∂ t U + A i ∂ x i U + = 0 , BU i =1 E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 5 / 32
Introduction Model Problems Two Problems Recast in the Time Domain Generalized hyperbolic system: Anisotropy � �� � Anisotropy+Dispersion d ���� � U ( x , t ) ∈ R m , A i , B ∈ R m × m , i = 1 , . . . , d . ∂ t U + A i ∂ x i U + = 0 , BU i =1 U → U 0 e i ( ω t − k · x ) = ⇒ F A , B ( ω, k ) = 0 = ⇒ ω : ℑ ω = 0 . E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 5 / 32
Introduction Model Problems Two Problems Recast in the Time Domain Generalized hyperbolic system: Anisotropy � �� � Anisotropy+Dispersion d ���� � U ( x , t ) ∈ R m , A i , B ∈ R m × m , i = 1 , . . . , d . ∂ t U + A i ∂ x i U + = 0 , BU i =1 U → U 0 e i ( ω t − k · x ) = ⇒ F A , B ( ω, k ) = 0 = ⇒ ω : ℑ ω = 0 . We rescale the equations so that c = 1 = ǫ 0 = µ 0 . ω c = 0 (isotropic dispersive) ∂ t E − curl B = − J , ∂ t B + curl E = 0 , ∂ t J = ω 2 p E . ω c → ∞ (anisotropic dispersive) ∂ t E − curl B = − j p e 3 , ∂ t B + curl E = 0 , ∂ t j p = ω 2 p E z . E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 5 / 32
Introduction Introduction into the PML Maxwell Equations in Plasma: Dealing with Unbounded Domains The PML approach deals with the unboundedness of the domain Physical Domain PML (non-physical medium) E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 6 / 32
Introduction Introduction into the PML Maxwell Equations in Plasma: Dealing with Unbounded Domains The PML approach deals with the unboundedness of the domain Physical Domain PML (non-physical medium) Properties of the PML ’Perfect matching’, i.e. zero reflections at the interface between the physical domain and the 1 PML E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 6 / 32
Introduction Introduction into the PML Maxwell Equations in Plasma: Dealing with Unbounded Domains The PML approach deals with the unboundedness of the domain Physical Domain PML (non-physical medium) Properties of the PML ’Perfect matching’, i.e. zero reflections at the interface between the physical domain and the 1 PML Inside the PML the solution decays exponentially fast, so that on the PML boundary zero 2 Dirichlet or Neumann BCs can be posed E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 6 / 32
Introduction Introduction into the PML Maxwell Equations in Plasma: Dealing with Unbounded Domains The PML approach deals with the unboundedness of the domain Physical Domain PML (non-physical medium) Properties of the PML ’Perfect matching’, i.e. zero reflections at the interface between the physical domain and the 1 PML Inside the PML the solution decays exponentially fast, so that on the PML boundary zero 2 Dirichlet or Neumann BCs can be posed Non-physical/anisotropic phenomenon 3 E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 6 / 32
Introduction Introduction into the PML Standard PML for Maxwell Equations B´ erenger 1994-1996: split formulation; Zhao and Cangellaris 1996: reinterpretation as a change of variables ( non-split formulation). E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 7 / 32
Introduction Introduction into the PML Standard PML for Maxwell Equations B´ erenger 1994-1996: split formulation; Zhao and Cangellaris 1996: reinterpretation as a change of variables ( non-split formulation). PML in the direction n n E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 7 / 32
Introduction Introduction into the PML Standard PML for Maxwell Equations B´ erenger 1994-1996: split formulation; Zhao and Cangellaris 1996: reinterpretation as a change of variables ( non-split formulation). PML in the direction n n e x One way to write a PML system (PML in e x -direction): E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 7 / 32
Introduction Introduction into the PML Standard PML for Maxwell Equations B´ erenger 1994-1996: split formulation; Zhao and Cangellaris 1996: reinterpretation as a change of variables ( non-split formulation). PML in the direction n n e x One way to write a PML system (PML in e x -direction): rewrite the equations in the frequency domain ( ∂ t → i ω ) 1 x Im x � 1 σ ( x ′ ) dx ′ , with perform a change of variables ˜ x := x + 2 i ω 0 σ ( x ′ ) > 0 for x > 0 (analytic continuation) Re x come back to the time-domain 3 E. B´ ecache, P. Joly, M. Kachanovska (POEMS, INRIA, ENSTA ParisTech) Perfectly Matched Layers for Maxwell Equations in Plasmas December 3, 2014 7 / 32
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