An inverse problem of electromagnetic shaping of 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 = µ 0 j 0 in Ω ∇ . B = 0 in Ω B .ν = 0 on ∂ω = Γ || B || → 0 at ∞ � B � 2 + σ H + ρ g · x 3 = p 0 on Γ 2 µ 0 µ 0 the magnetic permeability. the unit normal vector. ν σ the surface tension. H the mean curvature of Γ = ∂ Ω . p 0 a constant. j 0 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 = µ 0 j 0 in Ω ∇ . B = 0 in Ω Ω j 0 B .ν = 0 on ∂ω = Γ || B || → 0 at ∞ � B � 2 + σ C = p 0 on Γ 2 µ 0 ω Γ j 0 = ( 0 , 0 , j 0 ) is the current density. m � j 0 = I ( α p δ x p ) p = 1 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 = µ 0 j 0 in Ω and ∇ . B = 0 in Ω imply that there exist a potential function ϕ : Ω → R such that B = ( ϕ y , − ϕ x , 0 ) and ϕ is the solution of: − ∆ ϕ = µ 0 j 0 in Ω = 0 on Γ ϕ 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 � ||∇ ϕ || 2 + σ P ( ω ) 2 µ 0 Ω where P ( ω ) is the perimeter of ω = Ω c . 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 S 0 . 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 j 0 solution of the following optimization problem: min δ (Ω , Ω ∗ ) j 0 where Ω ∈ O the set of admissible domains, with the following constraints: − ∆ ϕ = µ 0 j 0 in Ω ϕ = 0 on ∂ Ω ϕ ( x ) = O ( 1 ) as || x || → ∞ 1 || ∂ϕ ∂ν || 2 + σ C = p 0 on Γ 2 µ 0 � dx = S 0 ω 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 : || V || 2 min L 2 (Γ ∗ ) j 0 with the following constraints: � ( 1 || ∂ϕ � ∂ν || 2 + σ C ) Z .ν d γ = p 0 Z .ν d Γ 2 µ 0 Γ Γ for all Z in C 1 ( R 2 , R 2 ) and − ∆ ϕ = µ 0 j 0 in Ω ϕ = 0 on Γ (1) ϕ ( x ) = O ( 1 ) as || x || → ∞ � dx = S 0 ω 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 of the problem: || P || 2 min L 2 (Γ ∗ ) j 0 such that: Γ ∗ ( 1 || ∂ϕ � � ∂ν || 2 + σ C + P ) Z .ν d Γ = Γ ∗ p 0 Z .ν d Γ ∀ Z ∈ C 1 ( R 2 , R 2 ) 2 µ 0 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 ! 3 x 10 5 4.5 4 Objective function 3.5 3 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 30 a) Iteration ! 7 x 10 1.4 1.2 1 Objective function 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 b) Iteration 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 ! 3 x 10 3.5 3 2.5 Objective function 2 1.5 1 0.5 0 0 50 100 150 200 250 300 350 400 a) Iteration ! 4 x 10 4.5 4 3.5 Objective function 3 2.5 2 1.5 1 0.5 0 0 50 100 150 200 b) Iteration Figure: Example 2, Evolution of the cost function during the Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals iterations.a) formulation 1, b) formulation 2
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 ! 3 x 10 8 7 6 Objective function 5 4 3 2 1 0 0 5 10 15 20 a) Iteration ! 7 x 10 2.5 2 Objective function 1.5 1 0.5 0 0 5 10 15 20 25 b) Iteration Figure: Example 1b, Evolution of the cost function during the Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals iterations.a) formulation 1, b) formulation 2.
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 0.014 0.012 0.01 Objective function 0.008 0.006 0.004 0.002 0 0 5 10 15 20 a) Iteration ! 7 x 10 9 8 7 Objective function 6 5 4 3 2 1 0 0 5 10 15 20 25 30 35 b) Iteration Figure: Example 2a, Evolution of the cost function during the Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals iterations.a) formulation 1, b) formulation 2.
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 0.035 0.03 0.025 Objective function 0.02 0.015 0.01 0.005 0 0 5 10 15 20 a) Iteration ! 6 x 10 1.4 1.2 1 Objective function 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 30 b) Iteration Figure: Example 2a, Evolution of the cost function during the Alfredo Canelas , Jean R. Roche , Jose Herskovits An inverse problem of electromagnetic shaping of liquid metals iterations.a) formulation 1, b) formulation 2.
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
Recommend
More recommend
