Rigorous approach to the derivation of 1D models for wave propagation in electrical networks Patrick Joly (INRIA) EPI POEMS (UMR CNRS-ENSTA-INRIA) e with G. Beck (Poems, INRIA) and S. Imperiale (M3DISIM, INRIA) Analysis and Numerics of Acoustic and Electromagnetic Problems RRICAM Workshop, Linz, October 2016 Patrick Joly POems (INRIA)
Motivation This is motivated by applications in non destructive testing (with CEA LIST) Non destructive testing of electric networks (example : SNCF) Electromagnetic waves Defects often appear at junctions Coaxial cables We need an efficient simulation tool, in time domain, based on a simplified 1D model that keeps in memory the complex 3D structure of the cable. 2 Patrick Joly POems (INRIA)
Electromagnetic waves in thin cables This is a subject that is surely considered as perfectly understood by electrical engineers (via modal analysis) R. Dautray J. L. Lions Mathematical Analysis, Numerical method in Science and Technology (1993). Z. Menachem, Wave propagation in a curved waveguide with arbitrary dielectric transverse profiles , (2007) Hyo J. Eom. Electromagnetic theory for boundary value problems , (2004). Surprisingly (I may be wrong), it does not seem to have been intensively studied from a rigorous mathematical point of view S. Imperiale, P . Joly Error estimates for 1D asymptotic models in coaxial cables with heterogeneous cross-section (2012). S. Imperiale, P . Joly, Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section , (2013) G. Beck, S. Imperiale, P . Joly, Mathematical modeling of multi-conductor cables , (2014). G. Beck, Modélisation et analyse de réseaux électriques, PhD thesis (2016). 3 Patrick Joly POems (INRIA)
Part%I% Reduced%models% for%a%junction%of%wires Patrick Joly POems (INRIA)
The%geometry%of% the%problem Patrick Joly POems (INRIA)
The reference domain (1) the cables n o x ` T + x ` 3 e ` , x ` ⇥ ⇤ 0 ≤ ` ≤ L Ω ` = O ` + T ∈ S ` 3 x ` 2 x ` S ` 3 x ` 1 O ` e ` x ` T = ( x ` 1 , x ` 2 ) 3 L + 1 semi-infinite cables will play a privileged ` = 0 role in our presentation 6 Patrick Joly POems (INRIA)
The reference domain (1) the cables n o x ` T + x ` 3 e ` , x ` ⇥ ⇤ 0 ≤ ` ≤ L Ω ` = O ` + T ∈ S ` 3 x ` 2 x ` S ` 3 x ` 1 O ` e ` x ` T = ( x ` 1 , x ` 2 ) 3 dielectric materials Coaxial cables : Perfect conductors is not simply connected S ` 6 Patrick Joly POems (INRIA)
The reference domain (2) the junction ∂ b K i x 3 b connected and bounded K L [ ∂ b K = ∂ b K i ∪ ∂ b K e ∪ S ` ∂ b K e x 2 ` =0 x 1 7 Patrick Joly POems (INRIA)
The reference domain (3) the whole domain L [ J = b b Ω ` K ∪ ` =0 ε , µ : b → R + J − ε ( x ` T , x ` 3 ) = ε ` ( x T ) µ ( x ` T , x ` 3 ) = µ ` ( x T ) → R + ε ` , µ ` : S ` − 8 Patrick Joly POems (INRIA)
A family of geometrical domains J δ = δ b J 3D scaling δ δ δ 9 Patrick Joly POems (INRIA)
A family of geometrical domains L J � = K � ∪ [ Ω � ` ` =1 δ ε � ( x ) = ε ` ( x ` T / δ ) δ µ � ( x ) = µ ` ( x ` T / δ ) O � ` n o Ω � O � δ x ` T + x ` 3 e ` , x ` ⇥ ⇤ T ∈ S ` ` = ` + 3 2D scaling (transverse) K δ = δ b K ε δ ∂ t E δ − rot H δ = 0 3D scaling J δ µ δ ∂ t H δ + rot E δ = 0 δ ε δ ( x ) = ε ( x/ δ ) ∂ J δ E δ × n = 0 µ δ ( x ) = µ ( x/ δ ) + (well prepared) initial conditions 9 Patrick Joly POems (INRIA)
A family of geometrical domains L L J � = K � ∪ [ Ω � Y G = [0 , + ∞ [ ` ` =1 ` =0 ε � ( x ) = ε ` ( x ` T / δ ) µ � ( x ) = µ ` ( x ` T / δ ) n o Ω � O � δ x ` T + x ` 3 e ` , x ` ⇥ ⇤ T ∈ S ` ` = ` + 3 2D scaling (transverse) K δ = δ b K Objective 1 : Propose an approximate model, posed 3D scaling with 1D unknowns on the limit graph Objective 2 : Propose a reconstruction of the 3D field ε δ ( x ) = ε ( x/ δ ) from the 1D unknowns µ δ ( x ) = µ ( x/ δ ) Objective 3 : Justify rigorously the approximate model via error estimates 9 Patrick Joly POems (INRIA)
A family of geometrical domains This is the 3D Maxwell counterpart of the work achieved for acoustics in the PhD thesis of A. Semin P . Joly, A. Semin Construction and analysis of improved Kirchhoff conditions in a junction of thin slots (2008). A. Semin Propagation d’ondes dans des jonctions de fentes minces (Université Paris Sud (2008). Objective 1 : Propose an approximate model, posed with 1D unknowns on the limit graph Objective 2 : Propose a reconstruction of the 3D field from the 1D unknowns Objective 3 : Justify rigorously the approximate model via error estimates 10 Patrick Joly POems (INRIA)
The%simplified% 1D%model Patrick Joly POems (INRIA)
The 1D approximate model e 2 3 , t ) : R + × R + → R V � ` ( x ` 3 , t ) , I � ` ( x ` e 1 I � V � ` : 1D electric potentials ` : 1D electric currents ∂ ` : longitudinal derivative G C ` ∂ t V � ` + ∂ ` I � ` = 0 x ` 3 > 0 , t > 0 L ` ∂ t I � ` + ∂ ` V � ` = 0 e 0 L X V � 0 (0) − V � Z ` ,k ∂ t I � ` (0) = δ k (0) ` =1 L X I � ` (0) = δ Y ∂ t V � 0 (0) ` =0 12 Patrick Joly POems (INRIA)
The 1D approximate model e 2 3 , t ) : R + × R + → R V � ` ( x ` 3 , t ) , I � ` ( x ` e 1 I � V � ` : 1D electric potentials ` : 1D electric currents ∂ ` : longitudinal derivative G C ` ∂ t V � ` + ∂ ` I � ` = 0 x ` 3 > 0 , t > 0 L ` ∂ t I � ` + ∂ ` V � ` = 0 e 0 L X V � 0 (0) − V � Z ` ,k ∂ t I � ` (0) = δ k (0) 0 Standard ` =1 L Kichhoff conditions X I � ` (0) = δ Y ∂ t V � 0 (0) 0 ` =0 12 Patrick Joly POems (INRIA)
The 1D approximate model e 2 3 , t ) : R + × R + → R V � ` ( x ` 3 , t ) , I � ` ( x ` e 1 I � V � ` : 1D electric potentials ` : 1D electric currents ∂ ` : longitudinal derivative G C ` ∂ t V � ` + ∂ ` I � ` = 0 x ` 3 > 0 , t > 0 L ` ∂ t I � ` + ∂ ` V � ` = 0 e 0 L X V � 0 (0) − V � Z ` ,k ∂ t I � ` (0) = δ k (0) Improved Kichhoff conditions ` =1 L X I � ` (0) = δ Y ∂ t V � 0 (0) Junction defects ` =0 13 Patrick Joly POems (INRIA)
The 1D approximate model C ` ∂ t V � ` + ∂ ` I � C ` > 0 , L ` > 0 ` = 0 x ` 3 > 0 , t > 0 L ` ∂ t I � ` + ∂ ` V � ` = 0 Y > 0 L X V � ` (0) − V � Z ` ,k ∂ t I � 0 (0) = δ k (0) � � Z := Z ` k ` =1 L symmetric L × L X I � ` (0) = δ Y ∂ t V � 0 (0) positive definite ` =0 Well-posedness and stability : there is conservation of the energy L 1 Z + δ ` | 2 + L ` | I � 0 (0) | 2 + ⇣ ` | 2 ⌘ ⇣ �⌘ X C ` | V � Y | V � Z I � 0 , I � � 0 2 2 e � ` =0 ` I � I � � � 0 := ` (0) 1 ≤ ` ≤ L 14 Patrick Joly POems (INRIA)
The 1D approximate model C ` ∂ t V � ` + ∂ ` I � C ` > 0 , L ` > 0 ` = 0 x ` 3 > 0 , t > 0 L ` ∂ t I � ` + ∂ ` V � ` = 0 Y > 0 L X V � ` (0) − V � Z ` ,k ∂ t I � 0 (0) = δ k (0) � � Z := Z ` k ` =1 L symmetric L × L X I � ` (0) = δ Y ∂ t V � 0 (0) positive definite ` =0 Well-posedness and stability : there is conservation of the energy L 1 Z + δ ` | 2 + L ` | I � 0 (0) | 2 + ⇣ ` | 2 ⌘ ⇣ �⌘ X C ` | V � Y | V � Z I � 0 , I � � 0 2 2 e � ` =0 ` I � I � � � 0 := ` (0) 1 ≤ ` ≤ L 14 Patrick Joly POems (INRIA)
The%effective%coefficients% %%%%%%%%%(i)%%%%%%%%%%and L ` C ` Patrick Joly POems (INRIA)
The 2D transverse electric potentials ⇣ ⌘ S ` ε ` r T ϕ ` S ` div T = 0 e ϕ ` ∂ S ext e = 0 ` ϕ ` ∂ S int e = 1 ` u ` ∈ L 2 � � � E ` := T ( S ` ) / rot T u ` = 0 , div T = 0 , u ` × n ` = 0 on ∂ S ` ε ` u ` Z e | 2 d σ ε ` | r T ϕ ` r T ϕ ` ⇥ ⇤ C ` = E ` = span e S ` 16 Patrick Joly POems (INRIA)
Topology of the reference domain L [ J = b b Ω ` K ∪ ` =1 b J is not simply connected 17 Patrick Joly POems (INRIA)
Topology of the reference domain For each ` 6 = 0 Ω ` plane and orthogonal to S ` Γ ` plane and orthogonal to S 0 Ω 0 18 Patrick Joly POems (INRIA)
Topology of the reference domain Γ ` L [ b J \ ∀ 1 ≤ m ≤ L, Γ ` ` =1 is simply connected Ω 0 19 Patrick Joly POems (INRIA)
Topology of the reference domain Ω ` γ ` S ` Γ ` γ 0 , ` S 0 Ω 0 20 Patrick Joly POems (INRIA)
The 2D transverse magnetic potentials ⇣ ⌘ S ` µ ` r T ψ ` S ` \ γ ` div T = 0 m ∂ n ψ ` m = 0 ∂ S ` γ ` µ ∂ ⌫ ψ ` ψ ` ⇥ ⇤ ⇥ ⇤ ` = 0 ` = 1 γ ` m m 21 Patrick Joly POems (INRIA)
The 2D transverse magnetic potentials ⇣ ⌘ S ` µ ` r T ψ ` S ` \ γ ` div T = 0 m ∂ n ψ ` m = 0 ∂ S ` γ ` µ ∂ ⌫ ψ ` ψ ` ⇥ ⇤ ⇥ ⇤ ` = 0 ` = 1 γ ` m m u ` ∈ L 2 � � � H ` := T ( S ` ) / rot T u ` = 0 , div T µ ` u ` = 0 , u ` · n ` = 0 on ∂ S ` 21 Patrick Joly POems (INRIA)
The 2D transverse magnetic potentials ⇣ ⌘ S ` µ ` r T ψ ` S ` \ γ ` div T = 0 m ∂ n ψ ` m = 0 ∂ S ` γ ` µ ∂ ⌫ ψ ` ψ ` ⇥ ⇤ ⇥ ⇤ ` = 0 ` = 1 γ ` m m u ` ∈ L 2 � � � H ` := T ( S ` ) / rot T u ` = 0 , div T µ ` u ` = 0 , u ` · n ` = 0 on ∂ S ` ⇥ e ⇤ r T ψ ` H ` = span m 21 Patrick Joly POems (INRIA)
Recommend
More recommend
