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

Transcription

1 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, Petrópolis - RJ, Brazil, 3 I.E.C.N., Université de Lorraine, CNRS, INRIA, B.P , Vandoeuvre lès Nancy, France {roche}@iecn.u-nancy.fr PICOF-2012

2 Outline Model Problem Example The shape optimization inverse problem Inverse problem formulation, topological approach Kohn - Vogelius criterion Topological derivative Numerical Algorithm Numerical results References

3 Model Problem in 2d ϕ = µ 0 j 0 in Ω ϕ = 0 on Γ ϕ(x) = O(1) as x 1 ϕ 2 2µ 0 ν + σc = p 0 on Γ 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 Θp and p=1 Ω j 0 dx = 0.

4 example

5 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 2µ 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 = 2µ 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).

6 Inverse Problem, topological approach The equilibrium equation in terms of the flux : 1 ϕ 2µ 0 n 2 + σc = p 0 on Γ. Calling p = 2µ 0 (p 0 σc), 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.

7 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 W0 1 (Ω) where: W 1 0 (Ω) = {u : ρ u L2 (Ω) and u L 2 (Ω)}, with ρ(x) = [ 1 + x 2 log(2 + x 2 )] 1.

8 Kohn-Vogelius criterion We introduce a shape functional based on the Kohn-Vogelius criterion, namely ψ(0) = J(φ) = 1 2 φ 2 L 2 (Γ) = 1 φ 2 dγ, 2 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 n = κ p d(j 0) on Γ, φ(x) = c + o(1) as x, Γ where d(j 0 ) = Γ 1 Ω µ 0 j 0 dx.

9 Topological derivative Let us consider that the domain Ω is subject to a non-smooth perturbation: Ω = Ω ɛ = Ω B ε ( 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 functions x D i T ψ( x), 1 i 2, are the topological first and second order derivatives of ψ at x.

10 Topological derivative The term f 1 (ε)dt 1 ψ + f 2(ε)DT 2 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.

11 The perturbation is characterized by changing the electric current distribution j 0 by a new one j ε : j ε = j 0 + αiχ Bε(ˆx), where B ε (ˆx) denotes a ball of radius ε, center ˆx and B ε (ˆx) Ω. I is a given current density value and α = ±1 is the sign of the current density in B ε (ˆx). In this way, the shape functional associated to the perturbed problem reads: ψ(ε) = J(φ ε ) = 1 φ 2 ε ds, (1) 2 where φ ε is unique solution in W0 1 (Ω) to the following problem: φ ε = µ 0 j ε in Ω, φ ε = κ p d(j ε ) on Γ, n (2) φ ε ds = 0. Γ Γ

12 Topological derivative Theorem The topological derivatives of the shape functional are DT 1 ψ(ˆx) = αµ 0I φf ds, Γ DT 2 ψ(ˆx) = 1 2 µ2 0 I2 f 2 ds. where the function f W0 1 (Ω) satisfies the following problem: f = (πε 2 ) 1 χ Bε(ˆx) in Ω, f n = Γ 1 on Γ, f ds = 0. Γ Γ

13 Numerical algorithm The proposed approach is to solve the optimization problem 1 min j0 2 φ 2 L 2 (Γ) ρ j0 2 dx where m j 0 = I α p χ Θp and 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 ) Ω Ω j 0 dx = 0 method of optimization: add a new small circular region of current density αi and center ˆx Θ 0 in order to decrease the objective function.

14 With the adoption of a level-set domain representation, Θ + = {x Θ, ψ + (x) < 0}, Θ = {x Θ, ψ (x) < 0}. Let EV (ˆx, 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 αi of radius ε and center ˆx, namely, EV (ˆx, ε, α) = f 1 (ε)d 1 T ψ(ˆx) + f 2(ε)D 2 T ψ(ˆx). 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., EV (ˆx, ε, α) > 0, ˆx Θ +, and α = 1, EV (ˆx, ε, α) > 0, ˆx Θ, and α = +1, EV (ˆx, ε, α) > 0, ˆx Θ 0, and α = ±1.

15 Let { g + EV (ˆx, ε, 1) if ˆx Θ (x) = +, EV (ˆx, ε, +1) if ˆx Θ 0 Θ, { g EV (ˆx, ε, +1) if ˆx Θ (x) =, EV (ˆx, ε, 1) if ˆx Θ 0 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., (Amstutz, Andrä) h(x) = sign(x) x β with β > 0.

16 Algorithm Given ψ + 0 and ψ 0 L2 (Θ) n N where t n [0, 1] Lemma { 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 ) Assume that ψ ψ 0 0. Then ψ+ n + ψ n 0 n N.

17 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.

18 Numerical results 10 3 Coarse mesh Fine mesh 10 4 Objective function Iteration Figure: Evolution of the objective function.

19 Numerical results Figure: Dashed line: target shape. Solution for a mesh of cells of size 0.02 with β = 3 and ρ = Black area: positive inductors, gray area: negative inductors, thin solid line: equilibrium shape.

20 Numerical results 10 3 Coarse mesh Fine mesh 10 4 Objective function Iteration Figure: Evolution of the objective function.

21 Numerical results Total current ρ=0 ρ=1.0e 12 ρ=1.0e Iteration Figure: Evolution of the total current considering different values of ρ.

22 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.

23 Numerical results Coarse mesh Fine mesh Objective function Iteration Figure: Evolution of the objective function.

24 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.

25 Numerical results 10 2 Coarse mesh Fine mesh Objective function Iteration Figure: Evolution of the objective function.

26 Numerical results (a) (b) Figure: Dashed line: target shape. Solution for a mesh of cells of size mple Solutions withfor β a= mesh 3. Black of cells area: of size 0.02 positive and β inductors, =3. (a)ex41b,(b) gray area: negative rea: positive inductors, inductors, thingray solid area: line: negative equilibrium inductors, shape. dashed line: target d line: equilibrium shape.

27 Numerical results Coarse mesh Fine mesh Objective function Iteration Figure 14: Example Ex42a and Ex42b. Evolution of the objective function for β = 3. Figure: Evolution of the objective function. A. Canela, 0.07 A. A. Novotny, Jean R. Roche, Inverse Electromagnetic Casting Problem

28 Numerical results !=0!=1.0e 12!=1.0e 11 Total current Iteration Figure 16: Example Ex42b. Evolution of the total current considering different values of ρ. Figure: Evolution of the total current considering different values of ρ Example 5 The target shape of this example is depicted in Fig. 17. The current density I =0.2, σ = and µ 0 =1.0. Two cases, named Ex5a

29 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 optimization for inverse electromagnetic casting problems. Inverse Problems in Science and Engineering, available online, 2011.Doi = / 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 : , A. Canelas, J. R. Roche, J. Herskovits, Inductor shape optimization for electromagnetic casting, Structural and Multidisciplinary Optimization, 39(6) : , A. Canelas, J. R. Roche, J. Herskovits, The inverse electromagnetic shaping problem, Structural and Multidisciplinary Optimization, 38(4) : , 2009.