Un problème inverse: l identification du profil de courant dans un Tokamak

Transcription

1 Un problème inverse: l identification du profil de courant dans un Tokamak Blaise Faugeras en collaboration avec J. Blum et C. Boulbe Université de Nice Sophia Antipolis Laboratoire J.-A. Dieudonné Nice, France INRIA, Mars 2009 BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

2 Outline 1 Introduction 2 Mathematical modelling of axisymmetric equilibrium 3 The inverse equilibrium problem 4 Numerical method 5 Conclusion BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

3 JET Tokamak BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

4 JET : vacuum vessel and plasma BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

5 Introduction Equilibrium of a plasma : a free boundary problem Equilibrium equation inside the plasma, in an axisymmetric configuration : Grad-Shafranov equation Right-hand side of this equation is a non-linear source : the toroidal component of the plasma current density j φ Aim of this work Perform the real-time identification of this non-linearity from experimental measurements BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

6 Mathematical modelling of the equilibrium Momentum equation : ρ( u t + u. u) + p = j B At the slow resistive diffusion time scale ρ( u t + u. u) p Equilibrium equations p = j B (Conservation of momentum).b = 0 (Conservation of B) B = µj (Ampere s law) + axisymmetric assumption => Grad-Shafranov equation BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

7 Magnetic surfaces p B = 0 and p j = 0 => Field lines and current lines are on isobaric surfaces (p = cst) = magnetic surfaces BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

8 Axisymmetric configuration B = 0, conservation of B Cylindrical coordinates (e r, e φ, e z ) B independent of φ B = (B r e r + B z e z ) + B φ e φ = B p + B φ Poloidal flux : ψ(r, z) = 1 Bds = 2π D r 0 B z rdr B p = 1 r ψ = 1 r ( ψ e φ) B. ψ = 0. ψ = cst on each magnetic surface. p = p(ψ) BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

9 B = µ 0 j, Ampere s law j = 0 j = j p + j φ. j p = 1 r ( ( f ) e φ ) µ 0 j. p = 0. f p = 0. f = f (ψ) diamagnetic function B φ = f r e φ j φ = ( ψ)e φ with. = r ( 1 µ 0 r. r ) + z ( 1 µ 0 r. z ) BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

10 Grad-Shafranov equation j B = p, equilibrium equation (j p + j φ e φ ) (B p + B φ e φ ) = 1 µ 0 r B 1 φ f + j φ ψ = p. r p = p (ψ) ψ and f = f (ψ) ψ Grad-Shafranov equation In the plasma ψ = rp (ψ) + 1 µ 0 r (ff )(ψ) In the vacuum ψ = 0 BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

11 Tokamak BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

12 Definition of the free plasma boundary Two cases : magnetic separatrix : hyperbolic line with an X-point (left) outermost flux line inside the limiter (right) BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

13 Equation for ψ(r, z) inside the vacuum vessel where ψ = λ[ r A( ψ) + R 0 R 0 r B( ψ)]χ Ωp ψ = h on Ω R 0 major radius of the torus p ( ψ) = λ 1 R 0 A( ψ) 1 (ff )( µ ψ) = λr 0 B( ψ), 0 ψ max ψ ψ Ω = ψ b max ψ [0, 1] in Ω p Ω χ Ωp is the characteristic function of Ω p in Ω BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

14 The inverse equilibrium problem : experimental measurements magnetic measurements on the vacuum vessel ψ(m i ) = g i and 1 r ψ n (N j) = h j on Ω, I p = interferometry and polarimetry on several chords n e (ψ) ψ n e (ψ)dl = α m, C m C m r n dl = β m Ω p j φ dx kinetic pressure motional Stark effect p(r, 0) = p d (r) f j (B r (M j ), B z (M j ), B φ (M j )) = γ j BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

15 BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

16 Statement of the inverse problem State equation ψ = λ[ r A( R ψ) + R 0 0 r B( ψ)]1 Ωp ψ = g in Ω on Ω Least square minimization with J(A, B, n e ) = J 0 + J 1 + J 2 + J ɛ J 0 = k w k 2(1 r J 1 = k w k 2( J 2 = k w 2 k ( ψ n (N k) h k ) 2 C k n e ( ψ) ψ r n dl α k) 2 C k n e ( ψ)dl β k ) 2 BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

17 Statement of the inverse problem (2) Tikhonov regularization 1 J ɛ = ɛ A (A (x)) 2 dx 0 1 +ɛ B (B (x)) 2 dx 0 +ɛ ne 1 0 (n e (x)) 2 dx Find A 0, B 0, n e0 such that : J(A 0, B 0, n e0 ) = inf J(A, B, n e ) BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

18 Numerical identification Finite element resolution Find ψ H 1 with ψ = g on Ω such that v H0 1 1, Ω µ 0 r ψ vdx = λ[ r A( ψ) + R 0 Ω p R 0 r B( ψ)]vdx with Fixed point A(x) = i a if i (x), B(ψ) = i b if i (x), u = (a i, b i ) Kψ = Y (ψ)u + g K modified stiffness matrix, u coefficients of A and B, g Dirichlet BC Direct solver : (ψ n, u) ψ n+1 ψ n+1 = K 1 [Y (ψ n )u + g] BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

19 Numerical identification (2) Least-square minimization J(u) = C(ψ)ψ d 2 + u T Au d : experimental measurements A : regularization terms Approximation J(u) = C(ψ n )ψ d 2 + u T Au, with ψ = K 1 [Y (ψ n )u + g] J(u) = C(ψ n )K 1 Y (ψ n )u + C(ψ n )K 1 g d 2 + u T Au = E n u F n 2 + u T Au Normal equation. Inverse solver : ψ n u (E nt E n + A)u = E nt F n BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

20 EQUINOX One equilibrium reconstruction First guess (ψ 0, u 0, λ 0, Ω 0 p) Direct and Inverse problem fixed-point iterations : ne identification (normal equation, ψ n ) Update λ n+1 = I p / Ω n p[ r R 0 A n ( ψ n ) + R 0 r Bn ( ψ n )]dx and scale DOFs (m = max a i, u 1 m u) Inverse solver : ψ n u n+1 Direct solver : (ψ n, u n+1 ) ψ n+1, Ω n+1 p Stopping condition ψn+1 ψ n ψ n < ɛ BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

21 EQUINOX Real-time Quasi-static approach. Maximum number of fixed-point iterations Cheap operations : K = LU and K 1 computed and stored once for all Normal equation : 10 basis functions small linear system Tikhonov regularization parameters unchanged Expensive operations : ( ) Cmag Update of matrices C = and Y (ψ) C polar (ψ) Computation of products C(ψ)K 1 and C(ψ)K 1 Y (ψ) BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

22 Noise free twin experiment. Magnetics only. Identified A and B BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

23 Definitions Current density averaged over magnetic surfaces r 0 < j(r, ψ) r The average < A > is defined as < A >= Safety factor q For one field line q = φ 2π. >= λa( ψ) + λ r 2 0 < r 2 > B( ψ) C ψ q( ψ) = 1 2π Adl / B p C ψ C ψ B φ rb p dl dl B p BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

24 Noise free twin experiment. Magnetics only. Mean current density and safety factor BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

25 Noisy twin experiment. Magnetics only. Identified A and B BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

26 Noisy twin experiment. Magnetics only. Mean current density and safety factor BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

27 Noisy twin experiment. Magnetics and polarimetry. Identified A and B BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

28 Noisy twin experiment. Magnetics and polarimetry. Mean current density and safety factor BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

29 Numerical Results : Tore Supra and JET characteristics ToreSupra Finite element mesh JET Number of triangles Number of nodes functions A and B Basis type Bspline Bspline Number of basis func. 8 8 Computation time (1.80GHz) One equilibrium 20 ms 60 ms Real-time requirement : 100 ms BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

30 Tore Supra. Magnetics and polarimetry. BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

31 Jet Magnetics only. BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

32 Jet Magnetics and polarimetry. BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33

33 Conclusion EQUINOX : Real-time identification of the current density and equilibrium reconstruction Makes possible future real-time control of the current profile BF (Université de Nice) Identification équilibre plasma INRIA, Mars / 33