Time Domain Analysis of a Transmission Problem Between The
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Time domain analysis of a transmission Transparent boundary conditions problem between the vacuum and a for wave propagation in fractal networks metamaterial in electromagnetism Patrick JOLY Conference in the honor of Abderrahmane Bendali Pau, December 2017 with Maxence CASSIER and Christophe HAZARD Joint work with Maryna Kachanosvska and Adrien Semin UMR CNRS-ENSTA-INRIA When I began my PhD in 1980, for me, most important algerian people where footballers, a lot of them were playing in France Rachid Mekhloufi Lakhdar Belloumi Rabah Madjer Salah Assad Mustapha Dahleb Going every week at Ecole Polytechnique, I realised that Algeria had also a good team of applied mathematicians, many them playing in France Kamel Hamdache Rachid Touzani A. Bendali Mohamed Amara Youcef Amirat The only problem that I ever had with this distingiuished person is when I wanted to send him an e-mail becasuse of the spelling of his first name Abderramane Bendali h Finding the good solution needs several iterations, which is ok for a numerical analyst). The main issue was to find where to put the h , a a paradox for a numerical analyst Time domain analysis of a transmission Transparent boundary conditions problem between the vacuum and a for wave propagation in fractal networks metamaterial in electromagnetism Patrick JOLY Conference in the honor of Abderrahmane Bendali Pau, December 2017 with Maxence CASSIER and Christophe HAZARD Joint work with Maryna Kachanosvska and Adrien Semin UMR CNRS-ENSTA-INRIA Introduction Example : the lung An application : propagation of acoustic waves in humans lungs (to detect crackles). - dyadic tree with 23 generations, - over than 8 million slots, - the geometry is «almost» self similar. Molding obtained by Ewald R. Wiebel, University of Berne, Switzerland Can be modeled mathematically as an infinite quasi-self similar tree B. Maury, D. Salort, C. Vannier, Trace theorems for trees, applications for the human lung, Network and Heterogeneous Media 1 (3), 469-500 (2009) Adrien SEMIN Propagation of acoustic waves in fractal networks 10 Introduction Wave propagation in a tree Goal : study the propagation of acoustic waves in a network of thin slots and particularly in infinite trees (seen as a limit case of a very large number of slots) 11 5 10 9 2 4 8 n + ! 1 3 7 j =1 2 6 1 5 4 3 2 1 n =0 n =1 n =2 n =3 By a tree, we mean a graph with the additional notion of branches and successive generations natural numbering of edges with two indices i (n, j) ⌘ Adrien SEMIN Propagation of acoustic waves in fractal networks 11 The retained 1D modelWave propagation in a tree Notation : s denotes a generalized abcissa along the tree + o µ : R µ(s)=µi along the edge n i T ! µ @2u @ ( µ @ u)=0 t − s s 1D wave equations + Continuity + Kirchoff conditions ` u = u , (i, j) µi @s ui =0 i j 8 2 B` i i ` i X2B 2 B` This model is justified by an asymptotic analysis of the 3D acoustic wave equation in a thin network ( δ 0 ) with ! homogeneous Neumann boundary conditions δ ⇠ δ 2 The transverse cross section of the thin slot Γ i is µ i δ . Adrien SEMIN Propagation of acoustic waves in fractal networks 12 NumericalA complex simulations phenomenon - simulations : reflections up to infinity Solution computer with many generations and brute force Zoom with amplified color scale (one week of computation) Adrien SEMIN Propagation of acoustic waves in fractal networks 13 Difficulties of the problem Major difficulty : Treat Anumerically general the geometry fact that the tree is infinite can be done under the assumption that after a certain generation, all the subtrees are self-similar @su = T 1 u Transparent DtN conditions at nodes @su = T 2 u Main questions How to characterize and compute How to discretize and implement the operators T m ? them (in time) ? @su = T 3 u Similar questions were adressed in the following article, as a part of a series of works devoted to elliptic boundary value problems in fractal domains Y. Achdou, C, Sabot, N. Tchou, Diffusion and propagation problems in some ramified domains with a fractal boundary,ESAIM : M2AN 40(4), 623-652 (2006) Adrien SEMIN Propagation of acoustic waves in fractal networks 14 A self-similar p-adyc tree Σ : a root segment of length ` The tree is bounded e2 p strictly contractant direct ⌫21,1 Example with p =2 similitudes s j of ratio ↵ j < 1 . and ↵ 1 = ↵2 ⌫e12⌫,22 ` e1⌫,11 The branches of the tree are e0,1 1 e1⌫,2 ⌫e2⌫,31 e0,1 = ⌃ e2,4 e = s (e ), 1 k p ⌫2 1,k k 0,1 n en+1,p(j 1)+k = sk(en,j), 1 j p , 1 k p − Self-similarity of the coefficients : ( ⌫ , , ⌫ ) > 0 such that 9 1 ··· p µ s (e ) = ⌫ µ(e ) k n,j k · n,j 2 Example : the human lung p =2, ↵1 = ↵2 0.85, ⌫k = ↵k Adrien SEMIN Propagation of acoustic waves' in fractal networks 15 The reference DtNThe operator reference DtN operator 2 µ @ u @ ( µ @ u )=0 along , u'(0,t)='(t) t ' − s s ' T + some condition ( C ) at infinity (to be made precise) 1 The Dirichlet-to-Neumann operator : ⇤'(t):=@su'(0,t) ' ` =1 (C ) Reference tree 1 ⇤' Adrien SEMIN Propagation of acoustic waves in fractal networks 16 The reference DtNThe operator reference DtN operator 2 µ @ u @ ( µ @ u )=0 along , u'(0,t)='(t) t ' − s s ' T + some condition ( C ) at infinity (to be made precise) 1 The Dirichlet-to-Neumann operator : ⇤'(t):=@su'(0,t) It is a convolution operator characterized by its Fourier Laplace symbol + : '(t) '(!), ! C ⇤')(!)=⇤(!) '(!) F ! 2 F m ! I + C+ 1 '(!)= '(t) ei!t dt e ! 0 C R Z − Adrien SEMIN Propagation of acoustic waves in fractal networks 17 The reference DtNThe operator reference DtN operator 2 µ @ u @ ( µ @ u )=0 along , u'(0,t)='(t) t ' − s s ' T + some condition ( C ) at infinity (to be made precise) 1 The Dirichlet-to-Neumann operator : ⇤'(t):=@su'(0,t) It is a convolution operator characterized by its Fourier Laplace symbol + : '(t) '(!), ! C ⇤')(!)=⇤(!) '(!) F ! 2 F @ ( µ @ u)+µ !2 u =0 along , u(0, !)=1 s s T + some condition ( C ) at infinity (to be made precise) 1 ⇤(!) = @su(0, !) symbolic notation : ⇤ ⇤(@ ) ⌘ t Adrien SEMIN Propagation of acoustic waves in fractal networks 18 Scaling of the referenceScaling DtN of operator the DtN operator 2 µ @ u @ ( µ @ u )=0 along , u'(0,t)='(t) t ' − s s ' T + some condition ( C ) at infinity (to be made precise) 1 The Dirichlet-to-Neumann operator : ⇤'(t):=@su'(0,t) ' s s/` ! ` 1 ⇤' 1 @s `− @s ! `! 1 ⇤(!) `− ⇤(!`) Adrien SEMIN Propagation of acoustic waves in fractal networks 19 The referenceThe DtN transparent operator boundary condition 2 µ @ u @ ( µ @ u )=0 along , u'(0,t)='(t) t ' − s s ' T + some condition ( C ) at infinity (to be made precise) 1 The Dirichlet-to-Neumann operator : ⇤'(t):=@su'(0,t) (u , ) end point of the cut tree 1 T1 ' Q µ0 @su = µq @suq µ1 q=1 X u q =1 ··· µ0 ··· q = Q Q ⇤' µQ µ0 @su = µq ⇤q(@t) u q=1 Q X 1 (uQ, Q) T = µ0− µq ⇤q(@t) T q=1 X Adrien SEMIN Propagation of acoustic waves in fractal networks 20 The reference DtNThe operator reference DtN operator 2 µ @ u @ ( µ @ u )=0 along , u'(0,t)='(t) t ' − s s ' T + some condition ( C ) at infinity (to be made precise) 1 The Dirichlet-to-Neumann operator : ⇤'(t):=@su'(0,t) It is a convolution operator characterized by its Fourier Laplace symbol + : '(t) '(!), ! C ⇤')(!)=⇤(!) '(!) F ! 2 F @ ( µ @ u)+µ !2 u =0 along , u(0, !)=1 s s T + some condition ( C ) at infinity (to be made precise) 1 ⇤(!) = @su(0, !) symbolic notation : ⇤ ⇤(@ ) ⌘ t Dirichlet or Neumann condition at are defined in a variational way 1 Adrien SEMIN Propagation of acoustic waves in fractal networks 21 WeightedWeighted Sobolev spacesSobolev on s.s. spaces p-adyc treein p-adycs s.s. trees pn Notation : µ f ds := µ n,j f ds n 0 j=1 en,j ZT X≥ X Z 1 2 2 2 Weighted broken H - norm : u H1 = µ @su ds + µ u ds k k µ | | | | ZT ZT Associated Sobolev spaces 1 0 2 Hµ( )= v C ( ) / v H1 < (for Neumann) T 2 T k k µ 1 H1 ( ) H1 ( ) = H1 ( ) µ T (for Dirichlet) µ,0 T µ,c T 1 1 where H ( )= v H ( ) such that N/v =0in N µ,c T 2 µ T 9 T\T Adrien SEMIN Propagation of acoustic waves in fractal networks 22 The DirichletDirichlet and Neumannand Neumann Helmhotz Helhmoltz problems problems Given ! / R , we define the Dirichlet and Neumann (at ) problems 2 1 ( ) Find u H 1 ( ) / u ( 0 )=1 , such that Pd 2 µ,0 T 2 1 µ u0v0 ! µ u v =0, v H ( ) such that v(0)=0 − 8 2 µ,0 T ZT ZT ( ) Find u H 1 ( ) / u ( 0 )=1 , such that Pn 2 µ T 2 1 µ u0v0 ! µ u v =0, v H ( ) such that v(0)=0 − 8 2 µ T ZT ZT For each ! / R , ( d ) ( resp. ( n ) ) admits a unique solution denoted 2 P P u (!, ) ( resp. u n ( ! , ) ) d · · Definition : ⇤ (!):=@ u (!, 0) ⇤ (!):=@ u (!, 0) d s Adriend SEMIN Propagation of acousticn waves in fractal networkss n 23 Mathematical analysisSome mathematical analysis 1 1 ⌫ ↵− := ⌫i ↵i− > ⌫ ↵ := ⌫i ↵i max ↵i < 1 1 i p h i h i X X ⌫ ↵ h i 2 2 Lµ( )= v :( ) C / µ v < + T T 7! ZT | | 1} 1 H1 ( ) = H1 ( ) µ T 6 µ,0 T u (!, ) = u (!, ) n · 6 d · lung c ⇢ H1 ( ) L2 ( ) µ T ⇢ µ T 1 ⌫ ↵− 1 Adrien SEMIN Propagation of acoustic waves in fractal hnetworks i 24 Mathematical analysisSome mathematical analysis 1 1 ⌫ ↵− := ⌫i ↵i− > ⌫ ↵ := ⌫i ↵i max ↵i < 1 1 i p h i h i X X ⌫ ↵ h i H1 ( )=H1 ( ) µ T µ,0 T u (!, ) u (!, ) n · ⌘ d · c H1 ( ) L2 ( ) 1 µ T ⇢ µ T H1 ( ) = H1 ( ) µ T 6 µ,0 T u (!, ) = u (!, ) n · 6 d · lung c ⇢ H1 ( ) L2 ( ) µ T ⇢ µ T 1 ⌫ ↵− 1 Adrien SEMIN Propagation of acoustic waves in fractal hnetworks i 25 Mathematical analysisSome mathematical analysis Definition : ⇤d(!):=@sud(!, 0) ⇤n(!):=@sun(!, 0) 1 Theorem : f a ( ! ):= ! − ⇤ a ( ! ) , a = d , n are Herglotz functions f is a Herglotz function f is analytic from C + into C+ () m ! m ! I I C+ f C+ e ! e ! R R T Time domain version