Level Set Solution of an Inverse Electromagnetic Casting Problem - PowerPoint PPT Presentation

Level Set Solution of an Inverse Electromagnetic Casting Problem using Topological Analysis A. Canelas 1 , A.A. Novotny 2 and J.R.Roche 3 1 Instituto de Estructuras y Transporte, Facultad de Ingeniería, UDELAR, J. Herrera y Reissig, CP 11300, Montevideo, Uruguay, 2 Laboratório Nacional de Computação Cientifífica LNCC/MCT, Av. Getúlio Vargas 333, 25651-075 Petrópolis - RJ, Brazil, 3 I.E.C.N., Université de Lorraine, CNRS, INRIA, B.P . 70239, 54506 Vandoeuvre lès Nancy, France {roche}@iecn.u-nancy.fr PICOF-2012 A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

Outline Model Problem Example The shape optimization inverse problem Inverse problem formulation, topological approach Kohn - Vogelius criterion Topological derivative Numerical Algorithm Numerical results References A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

Model Problem in 2d − ∆ ϕ = µ 0 j 0 in Ω ϕ = 0 on Γ ϕ ( x ) = O ( 1 ) as || x || → ∞ � � 2 � � 1 ∂ϕ � � + σ C = p 0 on Γ � � 2 µ 0 ∂ν j 0 = ( 0 , 0 , j 0 ) is the current density. P ( ω ) is the perimeter of ω = Ω c � m � j 0 = I α p χ Θ p and j 0 dx = 0 . Ω p = 1 A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

example A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

The shape optimization inverse problem In the two dimensional case, we assume ω simply connected, the boundary is only one Jordan curve Γ . We assume also that j 0 is compactly supported in Ω . If p 0 ≥ 2 µ 0 σ max x ∈ Γ C ( x ) there exist B = ( ϕ y , − ϕ x , 0 ) if and only if (i) Γ is a analytic curve. (ii) If p 0 = 2 µ 0 σ max x ∈ Γ C ( x ) , this global maximum must be attain in a number even of points. And the magnetic field is well determined in a neighborhood of ω (local uniqueness). See (Henrot and Pierre). A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

Inverse Problem, topological approach The equilibrium equation in terms of the flux : � � 2 � � 1 ∂ϕ � � + σ C = p 0 on Γ . � � 2 µ 0 ∂ n � Calling ¯ p = 2 µ 0 ( p 0 − σ C ) , with p 0 , σ and C known, the equilibrium constraint in terms of the flux function reads: ∂ϕ ∂ n = κ ¯ p on Γ , where κ = ± 1, with the sign changes located at points where the curvature of Γ is a global maximum. We have two possible ways to define κ . However, both lead to the same solution j 0 but with the opposite sign. A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

Inverse Problem, topological approach We formulate the inverse problem as follows: determine the electric current density j 0 and the real constant c such that the system  − ∆ ϕ = µ 0 j 0 in Ω ,     ϕ = 0 on Γ , ∂ϕ = κ ¯  p on Γ ,   ∂ n  ϕ ( x ) = c + o ( 1 ) as � x � → ∞ , has a solution ϕ ∈ W 1 0 (Ω) where: W 1 ρ u ∈ L 2 (Ω) and ∇ u ∈ L 2 (Ω) } , 0 (Ω) = { u : � 1 + � x � 2 log ( 2 + � x � 2 )] − 1 . with ρ ( x ) = [ A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

Kohn-Vogelius criterion We introduce a shape functional based on the Kohn-Vogelius criterion, namely � ψ ( 0 ) = J ( φ ) = 1 L 2 (Γ) = 1 2 � φ � 2 | φ | 2 d Γ , 2 Γ where the auxiliary function φ depends implicitly on j 0 and c by solving the following boundary-value problem  − ∆ φ = µ 0 j 0 in Ω ,   ∂φ = κ ¯ p − d ( j 0 ) on Γ ,  ∂ n  φ ( x ) = c + o ( 1 ) as � x � → ∞ , where � d ( j 0 ) = | Γ | − 1 µ 0 j 0 dx . Ω A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

Topological derivative Let us consider that the domain Ω is subject to a non-smooth perturbation: Ω ǫ = Ω � B ε ( � Ω = ⇒ x ) ψ ( 0 ) = ⇒ ψ ( ε ) Then, if the topologically perturbed shape functional ψ ( ε ) , admits the following topological asymptotic expansion: ψ ( ε ) = ψ ( 0 ) + f 1 ( ε ) D 1 T ψ + f 2 ( ε ) D 2 T ψ + o ( f 2 ( ε )) , where f i ( ε ) , 1 ≤ i ≤ 2, are positive functions such that f i ( ε ) → 0, and f 2 ( ε ) / f 1 ( ε ) → 0, when ε → 0, we say that the x �→ D i functions � T ψ ( � x ) , 1 ≤ i ≤ 2, are the topological first and second order derivatives of ψ at � x . A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

Topological derivative The term f 1 ( ε ) D 1 T ψ + f 2 ( ε ) D 2 T ψ can be seen as a second order correction of ψ ( 0 ) to approximate ψ ( ε ) . In fact, the topological derivatives are scalar functions defined over the original domain that indicate, at each point, the sensitivity of the shape functional when a singular perturbation of size ε is introduced at that point. A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

The perturbation is characterized by changing the electric current distribution j 0 by a new one j ε : j ε = j 0 + α I χ B ε (ˆ x ) , where B ε (ˆ x ) denotes a ball of radius ε , center ˆ x and B ε (ˆ x ) ⊂ Ω . I is a given current density value and α = ± 1 is the sign of the current density in B ε (ˆ x ) . In this way, the shape functional associated to the perturbed problem reads: � ψ ( ε ) = J ( φ ε ) = 1 φ 2 ε ds , (1) 2 Γ where φ ε is unique solution in W 1 0 (Ω) to the following problem:  − ∆ φ ε = µ 0 j ε in Ω ,     ∂φ ε = κ ¯ p − d ( j ε ) on Γ , (2) � ∂ n     φ ε ds = 0 . Γ A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

Topological derivative Theorem The topological derivatives of the shape functional are � D 1 T ψ (ˆ x ) = αµ 0 I φ f ds , Γ � x ) = 1 f 2 ds . D 2 2 µ 2 0 I 2 T ψ (ˆ Γ where the function f ∈ W 1 0 (Ω) satisfies the following problem:  = ( πε 2 ) − 1 χ B ε (ˆ − ∆ f in Ω ,  x )    ∂ f = −| Γ | − 1 on Γ , � ∂ n     f ds = 0 . Γ A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

Numerical algorithm The proposed approach is to solve the optimization problem � 1 L 2 (Γ) + 1 2 � φ � 2 j 2 2 ρ min 0 dx j 0 Ω where � m � j 0 = I α p χ Θ p and j 0 dx = 0 Ω p = 1 Let Θ ⊂ Ω ; Θ = Θ + ∪ Θ − ∪ Θ 0 compact. Θ + set with current density j 0 positive. Θ − set with current density j 0 negative. Θ 0 = Θ \ (Θ + ∪ Θ − ) method of optimization: add a new small circular region of x ∈ Θ 0 in order to decrease the current density α I and center ˆ objective function. A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

With the adoption of a level-set domain representation, Θ + = { x ∈ Θ , ψ + ( x ) < 0 } , Θ − = { x ∈ Θ , ψ − ( x ) < 0 } . Let EV (ˆ x , ε, α ) be the expected variation of the objective function of problem for a perturbation of j 0 consisting in a circular region of current density α I of radius ε and center ˆ x , namely, x , ε, α ) = f 1 ( ε ) D 1 x ) + f 2 ( ε ) D 2 EV (ˆ T ψ (ˆ T ψ (ˆ x ) . A sufficient condition of local optimality for the class of perturbations considered is that the expected variation of the objective function be positive, i.e., x ∈ Θ + , and α = − 1 , EV (ˆ ∀ ˆ x , ε, α ) > 0 , x ∈ Θ − , and α = + 1 , EV (ˆ ∀ ˆ x , ε, α ) > 0 , x ∈ Θ 0 , and α = ± 1 . EV (ˆ ∀ ˆ x , ε, α ) > 0 , A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

Let � − EV (ˆ x ∈ Θ + , if ˆ x , ε, − 1 ) g + ( x ) = x ∈ Θ 0 ∪ Θ − , EV (ˆ if ˆ x , ε, + 1 ) � − EV (ˆ x ∈ Θ − , if ˆ x , ε, + 1 ) g − ( x ) = x ∈ Θ 0 ∪ Θ + . EV (ˆ if ˆ x , ε, − 1 ) The sufficient conditions are satisfied if the following equivalence relations between the functions g + and g − and the level-set functions ψ + and ψ − hold ∃ τ + > 0 h ( g + ) = τ + ψ + , s.t. ∃ τ − > 0 h ( g − ) = τ − ψ − , s.t. where h : R → R must be an odd and strictly increasing function, e.g., h ( x ) = sign ( x ) | x | β with β > 0 . (Amstutz, Andrä) A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

Algorithm Given ψ + 0 ∈ L 2 (Θ) 0 and ψ − ∀ n ∈ N � ψ + n + 1 = ( 1 − t n ) ψ + n + t n h ( g + n ) ψ − n + 1 = ( 1 − t n ) ψ − n + t n h ( g − n ) where t n ∈ [ 0 , 1 ] Lemma Assume that ψ + 0 ≥ 0 . Then ψ + 0 + ψ − n + ψ − n ≥ 0 ∀ n ∈ N . A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

Numerical results Figure: Dashed line: target shape. Solution for a mesh of cells of size 0 . 02 with β = 3. Black area: positive inductors, gray area: negative inductors, thin solid line: equilibrium shape. A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem

Numerical results −3 10 Coarse mesh Fine mesh −4 10 Objective function −5 10 −6 10 −7 10 0 1 2 3 4 5 6 Iteration Figure: Evolution of the objective function. A. Canela, A. A. Novotny , Jean R. Roche, Inverse Electromagnetic Casting Problem