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Level Set Solution of an Inverse Electromagnetic Casting Problem using Topological Analysis

• A. Canelas1, A.A. Novotny2 and J.R.Roche3

1Instituto de Estructuras y Transporte, Facultad de Ingeniería, UDELAR, J.

Herrera y Reissig, CP 11300, Montevideo, Uruguay,

2Laboratório Nacional de Computação Cientifífica LNCC/MCT, Av. Getúlio Vargas

333, 25651-075 Petrópolis - RJ, Brazil,

3I.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 = µ0j0 in Ω ϕ = 0

• n Γ

ϕ(x) = O(1) as ||x|| → ∞ 1 2µ0

• ∂ϕ

∂ν

• 2

+ σC = p0 on Γ j0 = (0, 0, j0) is the current density. P(ω) is the perimeter of ω = Ωc j0 = I

m

• p=1

αpχΘp and

• Ω

j0 dx = 0 .

• 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 j0 is compactly supported in Ω. If p0 ≥ 2µ0σ max

x∈Γ C(x)

there exist B = (ϕy, −ϕx, 0) if and only if (i) Γ is a analytic curve. (ii) If p0 = 2µ0σ maxx∈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 : 1 2µ0

• ∂ϕ

∂n

• 2

+ σC = p0

• n Γ .

Calling ¯ p =

• 2µ0(p0 − σC), with p0, σ and C known, the

equilibrium constraint in terms of the flux function reads: ∂ϕ ∂n = κ ¯ p

• n 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 j0 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 j0 and the real constant c such that the system          −∆ϕ = µ0j0 in Ω , ϕ = 0

• n Γ ,

∂ϕ ∂n = κ ¯ p

• n Γ ,

ϕ(x) = c + o(1) as x → ∞ , has a solution ϕ ∈ W 1

0 (Ω) where:

W 1

0 (Ω) = {u :

ρ u ∈ L2(Ω) and ∇u ∈ L2(Ω)} , with ρ(x) = [

• 1 + x2 log(2 + x2)]−1.
• 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 2φ2

L2(Γ) = 1

2

• Γ

|φ|2dΓ , where the auxiliary function φ depends implicitly on j0 and c by solving the following boundary-value problem      −∆φ = µ0j0 in Ω , ∂φ ∂n = κ ¯ p − d(j0)

• n Γ ,

φ(x) = c + o(1) as x → ∞ , where d(j0) = |Γ|−1

• Ω

µ0j0 dx .

• 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 = ⇒ Ωǫ = Ω Bε( x) ψ(0) = ⇒ ψ(ε) Then, if the topologically perturbed shape functional ψ(ε), admits the following topological asymptotic expansion: ψ(ε) = ψ(0) + f1(ε)D1

Tψ + f2(ε)D2 Tψ + o(f2(ε)) ,

where fi(ε), 1 ≤ i ≤ 2, are positive functions such that fi(ε) → 0, and f2(ε)/f1(ε) → 0, when ε → 0, we say that the functions x → Di

Tψ(

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 f1(ε)D1

Tψ + f2(ε)D2 Tψ can be seen as a second order

correction of ψ(0) to approximate ψ(ε). In fact, the topological derivatives are scalar functions defined

• ver the original domain that indicate, at each point, the

sensitivity of the shape functional when a singular perturbation

• f 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 j0 by a new one jε : jε = j0 + αIχBε(ˆ

x) ,

where Bε(ˆ x) denotes a ball of radius ε, center ˆ x and Bε(ˆ x) ⊂ Ω. I is a given current density value and α = ±1 is the sign of the current density in Bε(ˆ x). In this way, the shape functional associated to the perturbed problem reads: ψ(ε) = J(φε) = 1 2

• Γ

φ2

ε ds ,

(1) where φε is unique solution in W 1

0 (Ω) to the following problem:

         −∆φε = µ0jε in Ω , ∂φε ∂n = κ ¯ p − d(jε)

• n Γ ,
• Γ

φε ds = 0 . (2)

• A. Canela, A. A. Novotny , Jean R. Roche,

Inverse Electromagnetic Casting Problem

Topological derivative

Theorem The topological derivatives of the shape functional are D1

Tψ(ˆ

x) = αµ0I

• Γ

φf ds , D2

Tψ(ˆ

x) = 1 2µ2

0I2

• Γ

f 2 ds . where the function f ∈ W 1

0 (Ω) satisfies the following problem:

         −∆f = (πε2)−1χBε(ˆ

x)

in Ω , ∂f ∂n = −|Γ|−1

• n Γ ,
• Γ

f ds = 0 .

• A. Canela, A. A. Novotny , Jean R. Roche,

Inverse Electromagnetic Casting Problem

Numerical algorithm

The proposed approach is to solve the optimization problem min

j0

1 2φ2

L2(Γ) + 1

2ρ

• Ω

j2

0 dx

where j0 = I

m

• p=1

αpχΘp and

• Ω

j0 dx = 0 Let Θ ⊂ Ω; Θ = Θ+ ∪ Θ− ∪ Θ0 compact. Θ+ set with current density j0 positive. Θ− set with current density j0 negative. Θ0 = Θ \ (Θ+ ∪ Θ−) method of optimization: add a new small circular region of current density αI and center ˆ x ∈ Θ0 in order to decrease the

• bjective 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 j0 consisting in a circular region of current density αI of radius ε and center ˆ x, namely, EV(ˆ x, ε, α) = f1(ε)D1

Tψ(ˆ

x) + f2(ε)D2

Tψ(ˆ

x) . A sufficient condition of local optimality for the class of perturbations considered is that the expected variation of the

• bjective function be positive, i.e.,

EV(ˆ x, ε, α) > 0 , ∀ˆ x ∈ Θ+ , and α = −1 , EV(ˆ x, ε, α) > 0 , ∀ˆ x ∈ Θ− , and α = +1 , EV(ˆ x, ε, α) > 0 , ∀ˆ x ∈ Θ0 , and α = ±1 .

• A. Canela, A. A. Novotny , Jean R. Roche,

Inverse Electromagnetic Casting Problem

Let g+(x) = −EV(ˆ x, ε, −1) if ˆ x ∈ Θ+ , EV(ˆ x, ε, +1) if ˆ x ∈ Θ0 ∪ Θ− , g−(x) = −EV(ˆ x, ε, +1) if ˆ x ∈ Θ− , EV(ˆ x, ε, −1) if ˆ x ∈ Θ0 ∪ Θ+ . 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 s.t. h(g+) = τ + ψ+ , ∃ τ − > 0 s.t. h(g−) = τ − ψ− , 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 and ψ− 0 ∈ L2(Θ)

∀n ∈ N ψ+

n+1 = (1 − tn)ψ+ n + tnh(g+ n )

ψ−

n+1 = (1 − tn)ψ− n + tnh(g− n )

where tn ∈ [0, 1] Lemma Assume that ψ+

0 + ψ− 0 ≥ 0 . Then 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

1 2 3 4 5 6 10

−7

10

−6

10

−5

10

−4

10

−3

Iteration Objective function Coarse mesh Fine mesh

Figure: Evolution of the objective function.

• 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 and ρ = 1.0 × 10−11. 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

10 20 30 40 50 10

−7

10

−6

10

−5

10

−4

10

−3

Iteration Objective function Coarse mesh Fine mesh

Figure: Evolution of the objective function.

• A. Canela, A. A. Novotny , Jean R. Roche,

Inverse Electromagnetic Casting Problem

Numerical results

10 20 30 40 50 −0.02 −0.01 0.01 0.02 0.03 0.04 Iteration Total current ρ=0 ρ=1.0e−12 ρ=1.0e−11

Figure: Evolution of the total current considering different values of ρ.

• 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 and 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

10 20 30 40 50 60 70 10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

Iteration Objective function Coarse mesh Fine mesh

Figure: Evolution of the objective function.

• 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

10 20 30 40 50 10

−5

10

−4

10

−3

10

−2

Iteration Objective function Coarse mesh Fine mesh

Figure: Evolution of the objective function.

• 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

50 100 150 200 10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

Iteration Objective function Coarse mesh Fine mesh

Figure: Evolution of the objective function.

• A. Canela, A. A. Novotny , Jean R. Roche,

Inverse Electromagnetic Casting Problem

Numerical results

50 100 150 200 −0.02 −0.01 0.01 0.02 0.03 0.04 Iteration Total current !=0 !=1.0e−12 !=1.0e−11

Figure: Evolution of the total current considering different values of ρ.

• A. Canela, A. A. Novotny , Jean R. Roche,

Inverse Electromagnetic Casting Problem

• A. Canelas, A.A. Novotny and J.R. Roche, Solution of

inverse electromagnetic casting problems using level-sets and second order topological derivatives, submitted to the Journal of Computational Physics,2012. J.R. Roche, A. Canelas and J. Herskovits, Shape

• ptimization for inverse electromagnetic casting problems.

Inverse Problems in Science and Engineering, available

• nline, 2011.Doi = 10.1080/17415977.2011.637206.
• A. Canelas, A.A. Novotny and J.R. Roche. A New Method

for Inverse Electromagnetic Casting Problems Based on the Topological Derivative. Journal of Computational Physics,230 : 3570 − 3588, 2011.

• A. Canelas, J. R. Roche, J. Herskovits, Inductor shape
• ptimization for electromagnetic casting, Structural and

Multidisciplinary Optimization, 39(6) : 589 − 606, 2009.

• A. Canelas, J. R. Roche, J. Herskovits, The inverse

electromagnetic shaping problem, Structural and Multidisciplinary Optimization, 38(4) : 389 − 403, 2009.

• A. Canela, A. A. Novotny , Jean R. Roche,

Inverse Electromagnetic Casting Problem