Inverse Eddy Current Problems

Transcription

1 Inverse Eddy Current Problems Bastian von Harrach (joint work with Lilian Arnold) Institut für Mathematik - IX, Universität Würzburg Oberwolfach workshop on Inverse Problems for Partial Differential Equations Oberwolfach, Germany, February 19 25, 2012.

2 Contents Motivation The direct problem The inverse problem

3 Motivation Inverse Electromagnetics & Eddy currents

4 Inverse Electromagnetics Inverse Electromagnetics: Generate EM field (drive excitation current through coil) Measure EM field (induced voltages in meas. coil) Gain information from measurements Applications: Metal detection (buried conductor) Non-destructive testing (crack in metal, metal in concrete)...

5 Maxwell s equations Classical Electromagnetics: Maxwell s equations curlh = ɛ t E +σe +J curle = µ t H in R 3 ]0,T[ in R 3 ]0,T[ E(x, t): Electric field H(x, t): Magnetic field J(x, t): Excitation current ɛ(x): Permittivity µ(x): Permeability σ(x): Conductivity Knowing J,σ,µ,ɛ + init. cond. determines E and H.

6 Eddy currents Maxwell s equations curlh = ɛ t E +σe +J curle = µ t H in R 3 ]0,T[ in R 3 ]0,T[ Eddy current approximation: Neglect displacement currents ɛ t E Justified for low-frequency excitations (Alonso 1999, Ammari/Buffa/Nédélec 2000) ( ) 1 t (σe)+curl µ curle = t J in R 3 ]0,T[

7 Where s Eddy? σ = 0: (Quasi-)Magnetostatics ( ) 1 curl µ curle = t J Excitation t J instantly generates magn. field 1 µ curle = th. σ 0: Eddy currents ( ) 1 t (σe)+curl µ curle = t J t J generates changing magn. field + currents inside conductor Induced currents oppose what created them (Lenz law)

8 Parabolic-elliptic equations ( ) 1 t (σe)+curl µ curle = t J in R 3 ]0,T[ parabolic inside conductor Ω = supp(σ) elliptic outside conductor Scalar example: (σu) t = u xx, u(,0) = 0, u x ( 2, ) = u x (2, ) = σ=0 σ=1 σ= σ=0 σ=1 σ= σ=0 σ=1 σ=

9 The direct problem Unified variational formulation for the parabolic-elliptic eddy current problem

10 Standard approach ( ) 1 t (σe)+curl µ curle = t J in R 3 ]0,T[ Standard approach: Decouple elliptic and parabolic part (e.g. Bossavit 1999, Acevedo/Meddahi/Rodriguez 2009) Find (E R 3 \Ω,E Ω ) H R 3 \Ω H Ω s.t. E Ω solves parabolic equation + init. cond. E R 3 \Ω solves elliptic equation interface conditions are satisfied Problem: Theory (solution spaces, coercivity constants, etc.) depends on Ω = suppσ and on lower bounds of σ Ω.

11 Unified approach? Parabolic-elliptic eddy current equation ( ) 1 t (σe)+curl µ curle = t J in R 3 ]0,T[ Inverse problem: Find σ (or Ω = suppσ) from measurements of E requires unified solution theory Test for unified theory: Can we linearize E w.r.t. σ? How does the solution of an elliptic equation change if the equation becomes a little bit parabolic? (For scalar analogue: Frühauf/H./Scherzer 2007, H. 2007)

12 Rigorous formulation Rigorous formulation: Let µ L +, σ L, σ 0, J t L 2 (0,T,W(curl) ) with divj t = 0 E 0 L 2 (R 3 ) 3 with div(σe 0 ) = 0. For E L 2 (0,T,W(curl)) the eddy current equations ( ) 1 t (σe)+curl µ curle = J t in R 3 ]0,T[ σe(x,0) = σ(x)e0 (x) in R 3 are well-defined and (if solvable) uniquely determine curle, σe.

13 Natural variational formulation Natural unified variational formulation (E 0 = 0 for simplicity): Find E L 2 (0,T,W(curl)) that solves T 0 R 3 (σe t Φ 1µ ) curle curlφ = for all smooth Φ with Φ(,T) = 0. T 0 J t Φ. R 3 equivalent to eddy current equation not coercive, does not yield existence results

14 Gauged formulation Gauged unified variational formulation (E 0 = 0 for simplicity) Find divergence-free E L 2 (0,T,W(curl)) that solves T 0 R 3 (σe t Φ 1µ ) curle curlφ = for all smooth divergence-free Φ with Φ(,T) = 0. T 0 J t Φ. R 3 coercive, yields existence and continuity results not equivalent to eddy current equation (σ const. divσe σdive) does not determine true solution up to gauge (curl-free) field

15 Coercive unified formulation How to obtain coercive + equivalent unified formulation? Ansatz E = A+ ϕ with divergence-free A. (almost the standard (A, ϕ)-formulation with Coulomb gauge) Consider ϕ = ϕ A as function of A by solving ( divσe = 0). divσ ϕ A = divσa. Obtain coercive formulation for A (Lions-Lax-Milgram Theorem Solvability and continuity results) A determines E (more precisely: curle and σe)

16 Unified variational formulation Unified variational formulation (Arnold/ H., SIAP, to appear) Find divergence-free A L 2 (0,T,W(curl)) that solves T 0 R 3 (σ(a+ ϕ A ) t Φ 1µ ) curla curlφ = for all smooth divergence-free Φ with Φ(,T) = 0. T 0 J t Φ. R 3 coercive, uniquely solvable E := A+ ϕ A is one solution of the eddy current equation curle, σe depend continuously on J t (uniformly w.r.t. σ) (for all solutions of the eddy current equation)

17 Solved and open problems Unified variational formulation allows to study inverse problems w.r.t. σ allows to rigorously linearize E w.r.t. σ around σ 0 = 0 (elliptic equation becoming a little bit parabolic in some region...) Open problem: Theory requires some regularity of Ω = suppσ and σ L +(Ω) in order to determine ϕ from A. Solution theory for for general σ L, σ 0? divσ ϕ = divσa

18 The inverse problem

19 Setup S Detecting conductors: Apply surface currents J on S (divergence-free, no electrostatic effects) Measure electric field E on S (tangential component, up to grad. fields) Measurement operator Ω Locate Ω = suppσ in t (σe)+curl Λ σ : J t γ τ E := (ν E S ) ν ( ) 1 µ curle = J t in R 3 ]0,T[ (+ zero IC) from all possible surface currents and measured values.

20 Measurement operator ν TL 2 := {u L 2 (S) 3 u ν = 0} S R 3 0 TL 2 := {u TL2 S u ψ = 0 smooth ψ} Measurement operator Λ σ : L 2 (0,T,TL 2 ) L 2 (0,T,TL 2 ), Jt γ τ E, where E solves eddy current eq. with [ν curle] S = J t on S. Remark TL 2 = TL 2 /TL 2 E not unique, but Λσ well-defined.

21 Sampling methods Working plan for the inverse problem: Linear Sampling Method (Colton/Kirsch 1996) characterizes subset of scatterer by range test allows fast numerical implementation Factorization Method (Kirsch 1998) characterizes scatterer by range test yields uniqueness under definiteness assumptions allows fast numerical implementation (FM for scalar parabolic-elliptic equ.: Frühauf/H./Scherzer 2007) Next step?

22 Sampling ingredients Ingredients for LSM and FM: Reference measurements: Λ := Λ σ Λ 0, Λ 0 : J t γ τ F, F solves curlcurlf = J t Time-integration: Consider I Λ, with I : E(, ) T 0 E(,t) dt Singular test functions G z,d (x) := curl d 4π x z, in R 3 ]0,T[. x R3 \{z}

23 LSM and FM Theorem (LSM): For every z below S and direction d R 3 γ τ G z,d R(IΛ) z Ω Theorem ( half FM): For every z below S and direction d R 3 γ τ G z,d R(I(Λ+Λ ) 1/2 ) z Ω Work in progress ( other half FM): holds.

24 Next step? Next step(?): Monotony methods For EIT: Λ σ NtD-operator for conductivity σ = 1+χ D D = Union of all balls B where Λ 1+χB Λ σ (H./Ullrich) (under the assumptions of the FM) stable test criterion (no infinity tests) allows fast numerical implementation allows extensions to indefinite cases

25 Conclusions Inverse transient eddy current problems require unified parabolic-elliptic theory can be approached by sampling methods (LSM/FM) Open problems Solution theory for for general σ L, σ 0? divσ ϕ = divσa Monotony based methods beyond EIT? Parabolic-elliptic problems? Inverse Scattering?