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 ( 0 ) with ! homogeneous Neumann boundary conditions ⇠ 2 The transverse cross section of the thin slot i is µ i .

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 : ⇤a'(t) @t'(t) dt 0 T>0, '(t) 8 8 Z0

Adrien SEMIN Propagation of acoustic waves in fractal networks 26 Mathematical analysisSome mathematical analysis

Definition : ⇤d(!):=@sud(!, 0) ⇤n(!):=@sun(!, 0)

1 Theorem : f a ( ! ):= ! ⇤ a ( ! ) , a = d , n are Herglotz functions

Theorem : ( ) and ( ) are well-posed except for a sequence of real Pd Pn n + n n + n frequencies ! d R , ! d + and ! n R , ! n + and ± 2 ⇤ ! 1 ± 2 ⇤ ! 1 n n ! ud(!, ) and u (!, ) are meromorphic, poles ! and ! ! · n · {± d } {± n }

Proof : spectral theory of self-adjoint operators with compact resolvent

m ! I

e ! R

Adrien SEMIN Propagation of acoustic waves in fractal networks 27 Mathematical analysisSome mathematical analysis

Definition : ⇤d(!):=@sud(!, 0) ⇤n(!):=@sun(!, 0)

1 Theorem : f a ( ! ):= ! ⇤ a ( ! ) , a = d , n are Herglotz functions

Theorem : ( ) and ( ) are well-posed except for a sequence of real Pd Pn n + n n + n frequencies ! d R , ! d + and ! n R , ! n + and ± 2 ⇤ ! 1 ± 2 ⇤ ! 1 n n ! ud(!, ) and u (!, ) are meromorphic, poles ! and ! ! · n · {± d } {± n }

+ 2 2 1 ⌦a,n How to compute Corollary : ⇤a(!)=⇤a ! a = d, n !2 !2 ⇤ a ( ! ) in practice ? n=0 a,n X One can make explicit as an integral time convolution operator t + 2 2 3 1 ⌦a,n ⇤ u = ⇤ u + K(0) @ u + K(t ⌧) @ u(⌧) d⌧ K(t):= cos !a,n(t) a a t t !2 Z0 n=0 a,n Adrien SEMIN Propagation of acoustic waves in fractal Xnetworks 28 Mathematical analysisSome mathematical analysis

Definition : ⇤d(!):=@sud(!, 0) ⇤n(!):=@sun(!, 0)

1 Theorem : f a ( ! ):= ! ⇤ a ( ! ) , a = d , n are Herglotz functions

Theorem : ( ) and ( ) are well-posed except for a sequence of real Pd Pn n + n n + n frequencies ! d R , ! d + and ! n R , ! n + and ± 2 ⇤ ! 1 ± 2 ⇤ ! 1 n n ! ud(!, ) and u (!, ) are meromorphic, poles ! and ! ! · n · {± d } {± n }

+ 2 2 1 ⌦a,n How to compute Corollary : ⇤a(!)=⇤a ! a = d, n !2 !2 ⇤ a ( ! ) in practice ? n=0 a,n X

This is where we shall really exploit the self-similar nature of the tree

Adrien SEMIN Propagation of acoustic waves in fractal networks 29 Scaling of the referenceScaling DtN of operator the DtN operator

2 @ u @ ( µ @ u )=0 along , u'(0,t)='(t) t ' s s ' T + some condition at infinity (to be made precise)

The Dirichlet-to-Neumann operator : ⇤'(t):=@su'(0,t)

'

` 1

⇤'

1 ⇤(!) ` ⇤(!`)

Adrien SEMIN Propagation of acoustic waves in fractal networks 30 A characterizationA characterization of the function of⇤ the function ⇤(!)

u1 1 2 @s( µ @su)+µ ! u =0 T T u u 2 2 u(0, !)=1 . T 0 1 . u0(0, !)=⇤(!)

{ up p T

2 2 ⇤(!) Along the first branch : @s u + ! u =0 u(1, !) = cos(!)+ sin(!) = ! ⇤(!) ) = u(s, !) = cos(! s)+ sin(! s) u0(1, !)=⇤(!) cos(!) ! sin(!) ) ! {

Kirchhoff condition at point 1 : u0(1, !)= ⌫i ui0 (1, !) X

1 Scaling argument (previous slide) : DtN for u0 (1, !)=↵ ⇤(↵j!) u(1, !) Tj j j

⌫i ⇤(!) ⇤(!) cos(!) ! sin(!)= cos(!)+ sin(!) ⇤(↵i !) ↵i ! X ⇣ ⌘ Adrien SEMIN Propagation of acoustic waves in fractal networks 31 A characterizationA characterization of the function of⇤ the function ⇤(!)

Both functions ⇤ d ( ! ) and ⇤ n ( ! ) solve the quadratic functional equation

⌫i ⇤(!) (E) ⇤(!) cos(!) ! sin(!)= ⇤(↵i!) cos(!)+ sin(!) ↵i ! ⇣ X ⌘⇣ ⌘ The frequency ! =0 plays a particular role : ⇤ = ⇤ (0) satisfies

⇤ = ⇤n := 0 ( u n (0) = 1 ) 1 ⇤ = ⌫↵ 1+⇤ ⇤ = () ) 1 1 1 h i ⇤ = ⇤d := ⌫↵ 1 ⌫↵ < 0 h i h i {

1 ⌫↵ < 1 h i

Adrien SEMIN Propagation of acoustic waves in fractal networks 32 A characterizationA characterization of the function of⇤ the function ⇤(!)

Both functions ⇤ d ( ! ) and ⇤ n ( ! ) solve the quadratic functional equation

⌫i ⇤(!) (E) ⇤(!) cos(!) ! sin(!)= ⇤(↵i!) cos(!)+ sin(!) ↵i ! ⇣ X ⌘⇣ ⌘ The frequency ! =0 plays a particular role : ⇤ = ⇤ (0) satisfies

⇤ = ⇤n := 0 ( u n (0) = 1 ) 1 ⇤ = ⌫↵ 1+⇤ ⇤ = () ) 1 1 1 h i ⇤ = ⇤d := ⌫↵ 1 ⌫↵ < 0 h i h i { Theorem: ⇤ d ( ! ) is the unique meromorphic function solution of (E) s. t.

⇤d(0) = ⇤d

⇤n(!) is the unique meromorphic function solution of (E) s. t.

⇤n(0) = ⇤n

Adrien SEMIN Propagation of acoustic waves in fractal networks 33 An algorithmAn algorithmfor the computation for the ofcomputation⇤ of ⇤(!)

⌫i ⇤(!) (E) ⇤(!) cos(!) ! sin(!)= ⇤(↵i!) cos(!)+ sin(!) ↵i ! ⇣ X ⌘⇣ ⌘

1 Setting f ( ! ):= ! ⇤ ( ! ) and f N ( ! ) = tan( ! ) , (E) can be rewritten as :

f b(!)+f N (!) f(!)= f b(!):= ⌫i f(↵i!) 1 f b(!) f N (!) X

(*) considering as an equation for ⇤ ( ! ) , assuming that the ⇤ ( ↵ i ! ) ’s are known

Adrien SEMIN Propagation of acoustic waves in fractal networks 34 An algorithmAn algorithmfor the computation for the ofcomputation⇤ of ⇤(!) 1 Setting f ( ! ):= ! ⇤ ( ! ) and f N ( ! ) = tan( ! ) , (E) can be rewritten as :

f b(!)+f N (!) f(!)= f b(!):= ⌫i f(↵i!) 1 f b(!) f N (!) X

+ + + Setting ↵ = max ↵ i < 1 and knowing f ( ! ) in B = z < C , this allows to compute + 1 + {| | } \ f ( ! ) in the larger domain ( ↵ ) B :

+ By induction, one can then determinate f ( ! ) in all C

The algorithm transmits (locally) the Herglotz property

+ One initiates by a Taylor expansion in B for small enough

2n Taylor expansion : Substituting ⇤ ( ! )= ⇤ 2 n ! in (E) allows to compute ⇤ 2 n by induction:

⇤2n = gn(⇤0, , ⇤2n 2X) with g n known explicitly ···

This is initiated with ⇤ 0 = ⇤ d or ⇤ 0 = ⇤ n which allows to distinct between Dirichlet and Neumann Adrien SEMIN Propagation of acoustic waves in fractal networks 35 An algorithmp =2 ↵for1 = the↵2 computation=0.4 µ1 = µof2 =0⇤ .4 Neumann case

Modulus

Imaginary part

Adrien SEMIN Propagation of acoustic waves in fractal networks 36 An algorithmp =2 ↵for1 = the↵2 computation=0.4 µ1 = µof2 =0⇤ .4 Neumann case

Modulus

Imaginary part

2 1 2 m ! ⇤(!) i ! sgn m !) C ! max 1, m ! 2 e |I | I Adrien SEMIN |Propagation| of acoustic waves in fractal|I networks| 37 The convolutionDiscretization: quadrature convolution approach quadrature approach

The continuous Dirichlet problem : ' := '(t) u' := u'(s, t)

2 µ @t u' @s( µ @su')=0 u'(0,t)='(t) +(C ) 1

t n t n ' := ' u' := u (s) The semi-discrete Dirichlet problem : { } { ' } n+1 n n 1 u 2u + u 1 ' ' ' n, 4 n n µ @s µ @su' =0 u'(0)=' +(C ) t2 1 n+1 n n 1 n, 1 u +2u + u with u 4 := ' ' ' (non-dissipative and unconditionally stable scheme) ' 4

t t n n The discrete DtN operator ⇤ : ⇤ ' := @su'(0) T Discrete positivity property = Stability⇤ '(t) @ 'of(t )thedt discrete0 DtN condition ) a t Z0

N 1 N 1 n+1 n n+1 n 1 n+1 n 1 u u 2 u + u 2 t n, 4 ' ' 1 ' ' ' ' ⇤ ' t = µ + @s t 2t 2 t 2 n=1 n=0 ZT n ⇣ ⌘ o X X Adrien SEMIN Propagation of acoustic waves in fractal networks 38 `The convolutionThe convolution quadrature approach quadrature approach

Discrete symbol : ⇤t ')(!)=⇤t (!) '(!) (discrete Fourier) F

t i 1 z i!t ⇤ (!)= ⇤ t 1+z z = e shift operator ⇣ ⌘ ⇣⌘

+ i 1 z + ! (unconditional stablility) C z < 1 t 1+z C 2 () | | () ⇣ ⌘ 2

n t m m t n q n q ⇤ (!) = (t) z = ⇤ ' = (t) ' ) q=0 X X i✓ Convolution weights : ( z = ⇢ e , ⇢ < 1 = Fourier series in ✓ ) ) m 2⇡ i✓ Fourier coefficients m ⇢ i 1 ⇢ e im✓ ⇢ < 1, (t)= ⇤ e d✓ 8 2⇡ t 1+⇢ ei✓ FFT algorithm Z0 ⇣ ⌘ Adrien SEMIN⇣⌘Propagation of acoustic waves in fractal networks 39 The convolution quadrature approach

Recap : transparent DtN operator at each end point e

1 Q e 2 1 1 0 . T = µ0 µq ⇤q(@t) ⇤q(!)=`q ⇤(`q !) . q=1 X Q

The fully discrete truncated tree problem:

n+1 n n 1 u 2u + u h n h v + µ @ un @ vn explicit t2 h s h s h ZTc ZTc 1 n, 4 implicit + Tt uh vh =0, e X2E

Q n, 1 n, 1 4 1 t 4 t 1 tq Tt u = µ0 µq ⇤q u ⇤q = `q ⇤ , tq = t/`q q=1 X Adrien SEMIN Propagation of acoustic waves in fractal networks 40 Numerical simulationsNumerical - simulations simulations

Neumann case

p =2 ` =2 ↵1 = ↵2 =0.4

h(t x) µ1 = µ2 =0.4

supp h(!) ! ⌦ ⇢ | | ⇢ max ⌦ 20 ⇡ max ⇡ T 0.1 min ⇡

0.1 min ⇡ Adrien SEMIN Propagation of acoustic waves in fractal networks 41 An algorithm for the Numericalcomputation ofsimulations⇤

Reference solution : computed with brute force (one week of computation) Approximate solution : computed with t =0.04 (Tmin 0.1) ⇡

1 Reference solution t =0.04 0.8

0.6 This part of the signals contains reflections from nodes which are 0.4 not in the computational domain

0.2

0

0.2 0 1 2 3 4 5 6 7 8 9 10

The convolution quadrature approach

Adrien SEMIN Propagation of acoustic waves in fractal networks 42 An algorithm for the Numericalcomputation ofsimulations⇤

Reference solution : computed with brute force (one week of computation) Approximate solution : computed with t =0.01 (Tmin 0.1) ⇡

1 Reference solution t =0.01 0.8

0.6 This part of the signals contains reflections from nodes which are 0.4 not in the truncated domain

0.2

0

0.2 0 1 2 3 4 5 6 7 8 9 10

The convolution quadrature approach

Adrien SEMIN Propagation of acoustic waves in fractal networks 43