An inverse problem of electromagnetic shaping of liquid metals - - PowerPoint PPT Presentation

An inverse problem of electromagnetic shaping

• f liquid metals

Alfredo Canelas 1 , Jean R. Roche 2 and Jose Herskovits 1

Mechanical Engineering Program - COPPE - Federal University of Rio de Janeiro, CT, Cidade Universiária, Ilha do Fundão, Rio de Janeiro, Brasil. {acanelas}@optimize.ufrj.br, {jose}@optimize.ufrj.br I.E.C.N., Nancy-Université, CNRS, INRIA B.P . 239, 54506 Vandoeuvre lès Nancy, France {roche}@iecn.u-nancy.fr

IMPA-2017

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Model Problem in 3d, the electromagnetic casting problem.

∇ × B = µ0j0 in Ω ∇ . B = in Ω B.ν =

• n ∂ω = Γ

||B|| → 0 at ∞ B2 2µ0 + σH + ρg · x3 = p0 on Γ µ0 the magnetic permeability. ν the unit normal vector. σ the surface tension. H the mean curvature of Γ = ∂Ω. p0 a constant. j0 is the current density.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

example

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Model Problem in 2d

∇ × B = µ0j0 in Ω ∇ . B = in Ω B.ν =

• n ∂ω = Γ

||B|| → 0 at ∞ B2 2µ0 + σC = p0 on Γ j0 = (0, 0, j0) is the current density. j0 = I (

m

• p=1

αpδxp)

ω Ω

j 0

Γ

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

The variational model of the direct problem

Conditions ∇ × B = µ0j0 in Ω and ∇ . B = 0 in Ω imply that there exist a potential function ϕ : Ω → R such that B = (ϕy, −ϕx, 0) and ϕ is the solution of: −∆ϕ = µ0j0 in Ω ϕ = 0

• n Γ

ϕ(x) = O(1) as ||x|| → ∞ Under suitable assumptions, the equilibrium configurations are given by a local critical point w.r.t. the domain of the following total energy: E(ω) = − 1 2µ0

• Ω

||∇ϕ||2 + σP(ω) where P(ω) is the perimeter of ω = Ωc. The variational formulation of the direct problem consists in considering the equilibrium domain ω as a stationary point for the total energy E(ω) under the constraint that measure of ω is given by S0.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

The shape optimization inverse problem in 2d

Given Ω∗ the target shape, we want to compute j0 solution of the following optimization problem: min

j0

δ(Ω, Ω∗) where Ω ∈ O the set of admissible domains, with the following constraints: −∆ϕ = µ0j0 in Ω ϕ =

• n ∂Ω

ϕ(x) = O(1) as ||x|| → ∞ 1 2µ0 ||∂ϕ ∂ν ||2 + σC = p0

• n Γ
• ω

dx = S0

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

First formulation of the inverse problem

Let V be a regular vector field with compact support in an open neighborhood of Ω∗ and Γ = (I + V)(Γ∗). Then the inverse problem formulation is the following : min

j0

||V||2

L2(Γ∗)

with the following constraints:

• Γ

( 1 2µ0 ||∂ϕ ∂ν ||2 + σC)Z.νdγ =

• Γ

p0Z.νdΓ for all Z in C1(R2, R2) and −∆ϕ = µ0j0 in Ω ϕ = 0

• n Γ

ϕ(x) = O(1) as ||x|| → ∞ (1)

• ω

dx = S0

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Second formulation, indirect approach

An indirect approach of the inverse problem can be considered if we introduce a slack variable function P(x) : Γ∗ → R in the equilibrium equation. Then we obtain the following formulation

• f the problem:

min

j0

||P||2

L2(Γ∗)

such that:

• Γ∗( 1

2µ0 ||∂ϕ ∂ν ||2+σC+P)Z.νdΓ =

• Γ∗ p0Z.νdΓ ∀ Z ∈ C1(R2, R2)

with the constraints (3). In this formulation the shape is no more a unknown of the problem.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

Test example: a) b)

Figure: Example 1,a) initial distribution of the inductors, b) final distribution of the inductors with formulation one and two.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

a)

5 10 15 20 25 30 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10

!3

Iteration Objective function

b)

5 10 15 20 25 0.2 0.4 0.6 0.8 1 1.2 1.4 x 10

!7

Iteration Objective function

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

Figure: Example 2 - Target shape and initial configuration of the inductors.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

a) b)

Figure: Example 2, final distribution of the inductors and final shape, a) formulation one, b) formulation two.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

Figure: Example 2 , Equilibrium shape obtained using he inductors resulting from the solution of the inverse problem by formulation two.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

a)

50 100 150 200 250 300 350 400 0.5 1 1.5 2 2.5 3 3.5 x 10

!3

Iteration Objective function

b)

50 100 150 200 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10

!4

Iteration Objective function

Figure: Example 2, Evolution of the cost function during the iterations.a) formulation 1, b) formulation 2

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

a) b)

Figure: Example 1b, a) initial distribution of the inductors, b) final distribution of the inductors.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

Figure: Example 1b , Equilibrium shape obtained using he inductors resulting from the solution of the inverse problem by formulation two.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

a)

5 10 15 20 1 2 3 4 5 6 7 8 x 10

!3

Iteration Objective function

b)

5 10 15 20 25 0.5 1 1.5 2 2.5 x 10

!7

Iteration Objective function

Figure: Example 1b, Evolution of the cost function during the iterations.a) formulation 1, b) formulation 2.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

a) b)

Figure: Example 2a, a) initial distribution of the inductors, b) final distribution of the inductors.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

a)

5 10 15 20 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Iteration Objective function

b)

5 10 15 20 25 30 35 1 2 3 4 5 6 7 8 9 x 10

!7

Iteration Objective function

Figure: Example 2a, Evolution of the cost function during the iterations.a) formulation 1, b) formulation 2.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

a) b)

Figure: Example 2b, a) initial distribution of the inductors, b) final distribution of the inductors.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

Figure: Example 2b , Equilibrium shape obtained using he inductors resulting from the solution of the inverse problem by formulation two.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

a)

5 10 15 20 0.005 0.01 0.015 0.02 0.025 0.03 0.035

Iteration Objective function

b)

5 10 15 20 25 30 0.2 0.4 0.6 0.8 1 1.2 1.4 x 10

!6

Iteration Objective function

Figure: Example 2a, Evolution of the cost function during the iterations.a) formulation 1, b) formulation 2.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

Figure: Example 6, Target shape and initial configuration of the inductors.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

a) b)

Figure: Example 6, final distribution of the inductors and final shape, a) formulation one, b) formulation two.

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

Figure: Example 6, Equilibrium shape obtained using he inductors resulting from the solution of the inverse problem by formulation two

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals

Numerical results

a)

50 100 150 200 250 300 350 400 0.02 0.04 0.06 0.08 0.1 0.12

Iteration Objective function

b)

50 100 150 200 250 300 350 400 0.5 1 1.5 2 2.5 3 3.5 4 x 10

!6

Iteration Objective function

Figure: Example 6, Evolution of the cost function during the iterations.a) formulation 1, b) formulation 2

Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals