Harrach Eddy

Direct and Inverse Eddy Current Problems Bastian von Harrach [email protected] (joint work with Lilian Arnold) Chair of Optimization and Inverse Problems, University of S tuttgart, Germany IFIP TC7.2 Workshop ”Electromagnetics - Modelling, Simulation, Control and Industrial Applications” Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany, May 13-17, 2013. B. Harrach: Direct and Inverse Eddy Current Problems Contents ◮Motivation ◮The direct problem ◮The inverse problem B. Harrach: Direct and Inverse Eddy Current Problems Motivation Inverse Electromagnetics & Eddy currents B. Harrach: Direct and Inverse Eddy Current Problems 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) ◮... B. Harrach: Direct and Inverse Eddy Current Problems Maxwell’s equations Classical Electromagnetics: Maxwell’s equations curlH=ǫ∂tE+σE+J inR3×]0,T[ curlE=−µ∂tH inR3×]0,T[ E(x,t): Electric field ǫ(x): Permittivity H(x,t): Magnetic field µ(x): Permeability J(x,t): Excitation current σ(x): Conductivity Knowing J,σ,µ,ǫ+ init. cond. determines E and H. B. Harrach: Direct and Inverse Eddy Current Problems Eddy currents Maxwell’s equations curlH=ǫ∂tE+σE+J inR3×]0,T[ curlE=−µ∂tH inR3×]0,T[ Eddy current approximation: Neglect displacement currents ǫ∂tE ◮Justified for low-frequency excitations (Alonso 1999, Ammari/Buffa/N´ ed´ elec 2000) /squiggleright∂t(σE)+curl/parenleftbigg1 µcurlE/parenrightbigg =−∂tJinR3×]0,T[ B. Harrach: Direct and Inverse Eddy Current Problems Where’s Eddy? ◮σ= 0:(Quasi-)Magnetostatics curl/parenleftbigg1 µcurlE/parenrightbigg =−∂tJ Excitation∂tJinstantly generates magn. field1 µcurlE=−∂tH. ◮σ/\e}atio\slash= 0:Eddy currents ∂t(σE)+curl/parenleftbigg1 µcurlE/parenrightbigg =−∂tJ ∂tJgenerates changing magn. field + currents inside conductor Induced currents oppose what created them (Lenz law) B. Harrach: Direct and Inverse Eddy Current Problems Parabolic-elliptic equations ∂t(σE)+curl/parenleftbigg1 µcurlE/parenrightbigg =−∂tJinR3×]0,T[ ◮parabolic inside conductor Ω = supp( σ) ◮ellipticoutside conductor Scalar example: ( σu)t=uxx,u(·,0) = 0,ux(−2,·) =ux(2,·) = 1. −2 −1 0 1 2−2−1012 σ=0 σ=1 σ=0 −2 −1 0 1 2−2−1012 σ=0 σ=1 σ=0 −2 −1 0 1 2−2−1012 σ=0 σ=1 σ=0 B. Harrach: Direct and Inverse Eddy Current Problems The direct problem Unified variational formulation for the parabolic-elliptic eddy current problem B. Harrach: Direct and Inverse Eddy Current Problems Standard approach ∂t(σE)+curl/parenleftbigg1 µcurlE/parenrightbigg =−∂tJinR3×]0,T[ Standard approach: Decouple elliptic and parabolic part (e.g. Bossavit 1999, Acevedo/Meddahi/Rodriguez 2009) Find (ER3\Ω,EΩ)∈HR3\Ω×HΩs.t. ◮EΩsolves parabolic equation + init. cond. ◮ER3\Ωsolves elliptic equation ◮interface conditions are satisfied Problem: Theory (solution spaces, coercivity constants, etc.) depends on Ω = supp σand on lower bounds of σ|Ω. B. Harrach: Direct and Inverse Eddy Current Problems Unified approach? Parabolic-elliptic eddy current equation ∂t(σE)+curl/parenleftbigg1 µcurlE/parenrightbigg =−∂tJinR3×]0,T[ Inverse problem : Findσ(or Ω = supp σ) from measurements of E ◮requiresunifiedsolution theory Test for unified theory: Can we linearize Ew.r.t.σ? How does the solution of an elliptic equation change if the equation becomes a little bit parabolic? (For scalar analogue: Fr¨ uhauf/ H./Scherzer 2007, H.2007) B. Harrach: Direct and Inverse Eddy Current Problems Rigorous formulation Rigorous formulation: Letµ∈L∞ +,σ∈L∞,σ≥0, Jt∈L2(0,T,W(curl)′) with div Jt= 0 E0∈L2(R3)3with div(σE0) = 0. ForE∈L2(0,T,W(curl)) the eddy current equations ∂t(σE)+curl/parenleftbigg1 µcurlE/parenrightbigg =−Jt inR3×]0,T[ √σE(x,0) =/radicalbig σ(x)E0(x) inR3 are well-defined and (if solvable) uniquely determine curl E,√ σE. B. Harrach: Direct and Inverse Eddy Current Problems Natural variational formulation Natural unified variational formulation ( E0= 0 for simplicity): FindE∈L2(0,T,W(curl)) that solves /integraldisplayT 0/integraldisplay R3/parenleftbigg σE·∂tΦ−1 µcurlE·curlΦ/parenrightbigg =/integraldisplayT 0/integraldisplay R3Jt·Φ. for all smooth Φ with Φ( ·,T) = 0. ◮equivalent to eddy current equation ◮not coercive , does not yield existence results B. Harrach: Direct and Inverse Eddy Current Problems Gauged formulation Gauged unified variational formulation ( E0= 0 for simplicity) Finddivergence-free E∈L2(0,T,W1(R3)) that solves /integraldisplayT 0/integraldisplay R3/parenleftbigg σE·∂tΦ−1 µcurlE·curlΦ/parenrightbigg =/integraldisplayT 0/integraldisplay R3Jt·Φ. for all smooth divergence-free Φ with Φ( ·,T) = 0. ◮coercive, yields existence and continuity results ◮not equivalent to eddy current equation (σ/\e}atio\slash= const./squigglerightdivσE/\e}atio\slash=σdivE) ◮doesnot determine true solution up to gauge (curl-free) field B. Harrach: Direct and Inverse Eddy Current Problems Coercive unified formulation How to obtain coercive + equivalent unified formulation? ◮AnsatzE=A+∇ϕwith divergence-free A. (almost the standard (A,ϕ)-formulation with Coulomb gauge) ◮Consider ∇ϕ=∇ϕAas function of Aby solving divσ∇ϕA=−divσA. (/squigglerightdivσE= 0). ◮Obtain coercive formulation for A (Lions-Lax-Milgram Theorem /squigglerightSolvability and continuity results) ◮Adetermines E (more precisely: curl Eand√σE) B. Harrach: Direct and Inverse Eddy Current Problems Unified variational formulation Unified variational formulation (Arnold/ H., SIAP, 2012) Finddivergence-free A∈L2(0,T,W1(R3)) that solves /integraldisplayT 0/integraldisplay R3/parenleftbigg σ(A+∇ϕA)·∂tΦ−1 µcurlA·curlΦ/parenrightbigg =/integraldisplayT 0/integraldisplay R3Jt·Φ. for all smooth divergence-free Φ with Φ( ·,T) = 0. ◮coercive, uniquely solvable ◮E:=A+∇ϕAisonesolution of the eddy current equation /squigglerightcurlE,√σEdepend continuously on Jt(uniformly w.r.t. σ) (forallsolutions of the eddy current equation) B. Harrach: Direct and Inverse Eddy Current Problems Asymptotic results Unified variational formulation ◮allows to rigorously linearize Ew.r.t.σaroundσ0= 0 (elliptic equation becoming a little bit parabolic in some r egion...) ◮easily extends from R3to bounded domain O (Osimply conn. with Lipschitz-boundary, ν∧E|∂O= 0) ◮justifiesparabolic regularization : IfEǫsolves ∂t(σǫEǫ)+curl/parenleftbigg1 µcurlEǫ/parenrightbigg =−∂tJinO×]0,T[, withσǫ(x) = max{σ(x),ǫ}then σǫEǫ→σE,curlEǫ→curlE (Arnold/ H., submitted to proceedings of IPDO 2013) B. Harrach: Direct and Inverse Eddy Current Problems Open problems ◮Theory requires some regularity of Ω = supp σandσ∈L∞ +(Ω) in order to determine ϕfromA. ◮Solution theory for divσ∇ϕ=−divσA for generalσ∈L∞,σ≥0? ◮Elliptic regularization of the variational formulation (i.e., adding ǫ/integraltextT 0/integraltext R3A·Φdx) is justified, but relation to elliptic regularization of the PDE ∂t(σE)+curl/parenleftbigg1 µcurlE/parenrightbigg +ǫE=−∂tJinR3×]0,T[, is not clear. B. Harrach: Direct and Inverse Eddy Current Problems The inverse problem B. Harrach: Direct and Inverse Eddy Current Problems Setup S ΩDetecting conductors: ◮Apply surface currents JonS (divergence-free, no electrostatic effects) ◮Measure electric field EonS (tangential component, up to grad. fields) ◮Measurement operator Λσ:Jt/ma√sto→γτE:= (ν∧E|S)∧ν Locate Ω = supp σin ∂t(σE)+curl/parenleftbigg1 µcurlE/parenrightbigg =−JtinR3×]0,T[ (+ zero IC) from all possible surface currents and measured values. B. Harrach: Direct and Inverse Eddy Current Problems Measurement operator S⊂R3 0ν TL2:={u∈L2(S)3|u·ν= 0} TL2 ⋄:={u∈TL2|/integraltext Su·∇ψ= 0 ∀smoothψ} Measurement operator Λσ:L2(0,T,TL2 ⋄)→L2(0,T,TL2 ⋄′),Jt/ma√sto→γτE, whereEsolves eddy current eq. with [ ν×curlE]S=JtonS. Remark TL2 ⋄′∼=TL2/TL2 ⋄⊥/squigglerightEnot unique, but Λ σwell-defined. B. Harrach: Direct and Inverse Eddy Current Problems Sampling methods Non-iterative shape detection methods: ◮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 ◮Beyond LSM/FM? B. Harrach: Direct and Inverse Eddy Current Problems Sampling ingredients Ingredients for LSM and FM: ◮Reference measurements : Λ := Λ σ−Λ0, Λ0:Jt/ma√sto→γτF,Fsolves curlcurl F=−JtinR3×]0,T[. ◮Time-integration : Consider IΛ, withI:E(·,·)/ma√sto→/integraltextT 0E(·,t)dt ◮Singular test functions Gz,d(x) := curld 4π|x−z|,x∈R3\{z} B. Harrach: Direct and Inverse Eddy Current Problems LSM and FM Arnold/H.(submitted) : For every zbelowS,z/\e}atio\slash∈Ω and direction d∈R3. Theorem (LSM) γτGz,d∈ R(IΛ)⇒z∈Ω Theorem (FM) If, additionally, sup µ|Ω<1 (diamagnetic scatterer) γτGz,d∈ R(I(Λ+Λ′)1/2)⇔z∈Ω B. Harrach: Direct and Inverse Eddy Current Problems Beyond LSM/FM? ◮Beyond LSM/FM?: Monotony methods ◮For EIT: Λ σNtD-operator for conductivity σ= 1+χD D= Union of all balls Bwhere Λ 1+χB≤Λσ(H./Ullrich) (under the assumptions of the FM) ◮stable test criterion (no infinity tests) ◮allows fast numerical implementation ◮allows extensions to indefinite cases B. Harrach: Direct and Inverse Eddy Current Problems Conclusions Inverse transient eddy current problems ◮require unified parabolic-elliptic theory ◮can be approached by sampling methods (LSM/FM) Open problems ◮Solution theory for divσ∇ϕ=−divσA for generalσ∈L∞,σ≥0? ◮Monotony based methods beyond EIT? Monotony for parabolic-elliptic problems? B. Harrach: Direct and Inverse Eddy Current Problems