Equilibrium reconstruction from discrete magnetic measurements in a - PowerPoint PPT Presentation

Equilibrium reconstruction from discrete magnetic measurements in a Tokamak Blaise Faugeras Jacques Blum and C´ edric Boulbe Universit´ e de Nice Sophia Antipolis Laboratoire J.-A. Dieudonn´ e Nice, France Blaise.Faugeras@unice.fr PICOF, April 2012 B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 1 / 19

Introduction. JET : vacuum vessel and plasma B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 2 / 19

Introduction. Tokamak B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 3 / 19

Introduction Equilibrium of a plasma : a free boundary problem Equilibrium equation inside the plasma, in axisymmetric configuration : Grad-Shafranov equation Right-hand side of this equation is a non-linear source : the toroidal component of the plasma current density Goal Identification of this non-linearity from experimental measurements. Perform the reconstruction of 2D equilibrium and the identification of the current density in real-time. B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 4 / 19

Mathematical modelling of the equilibrium 3D equilibrium equations  ∇ p = j × B (Conservation of momentum)  ∇ . B = 0 (Conservation of B )  ∇ × B = µ j (Ampere’s law) Axisymmetric assumption = > Grad-Shafranov equation 2D problem. Cylindrical coordinates ( r , φ, z ) State variable ψ ( r , z ) poloidal magnetic flux B p = 1 r ∇ ψ ⊥ B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 5 / 19

In the plasma : Grad-Shafranov equation − ∆ ∗ ψ := ∂ ∂ r ( 1 ∂ψ ∂ r ) + ∂ ∂ z ( 1 ∂ψ 1 ∂ z ) = rp ′ ( ψ ) + µ 0 r ( ff ′ )( ψ ) µ 0 r µ 0 r In the vacuum − ∆ ∗ ψ = 0 Boundary value problem  1  µ 0 r ff ′ ( ψ )]1 Ω p ( ψ ) − ∆ ∗ ψ = [ rp ′ ( ψ ) + Ω in  ψ = g Γ on B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 6 / 19

Definition of the free plasma boundary Two cases outermost flux line inside the limiter (left) magnetic separatrix : hyperbolic line with an X-point (right) B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 7 / 19

b Computational domain z − ∆ ∗ ψ = 0 Ω C i limiter ⊗ ⊗ Γ ⊗ ⊗ − ∆ ∗ ψ = j ( r, ψ ) C 0 r Ω p − ∆ ∗ ψ = j i ⊗ ⊗ flux loop Ω ⊗ ⊗ Ω 0 B probe B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 8 / 19

Inverse problem Step 1 : From discrete magnetic measurements to Cauchy conditions on a fixed contour Γ Magnetic measurements Flux loops : ψ ( M i ) B probes : B p ( N i ) . d i Cauchy conditions ( ψ , ∂ n ψ ) on Γ = ∂ Ω Dirichlet BC : direct problem Neumann BC : inverse problem Numerical methods Direct Interpolation (TCV EPFL, ToreSupra CEA Cadarache) Reconstruction of ψ in the vacuum - plasma boundary identification ◮ JET : ∆ ∗ ψ = 0, ψ piecewise polynomial ◮ Toroidal harmonics + PF coils current filaments model B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 9 / 19

Explicit solutions to ∆ ∗ ψ = 0 : toroidal harmonics Laplacian in cylindrical coordinates If ∆ ∗ ψ ( r , z ) = 0 in D then Ψ( r , z , φ ) = 1 r ψ ( r , z ) cos φ satisfies ∆Ψ = 0 in D × [0 2 π ] Quasi-separable solutions in bipolar (toroidal) coordinates � � Ψ( τ, η, φ ) = cosh τ − cosh η A ( τ ) B ( η ) cos φ Complete set of solutions � � P 1 � � � � cos( k η ) � � 2 (cosh τ ) a sinh τ k − 1 T P , Q √ cosh τ − cos η k ∈ N = Q 1 c , s , k 2 (cosh τ ) sin( k η ) k − 1 k ∈ N J. Segura and A. Gil. Evaluation of toroidal harmonics . CPC. 1999 Y. Fischer. PhD. 2011 B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 10 / 19

Flux in the vaccum N � � ( β P , Q c , s , k )( T P , Q ψ ( r , z ) = c , s , k ) + ψ f ( r , z ; r k , z k ) k =0 k N � � ( β P , Q B ( r , z ) = c , s , k ) B k ( r , z ) + B f ( r , z ; r k , z k ) k =0 k PF coils modelized by filaments of current Current I k at ( r k , z k ) : √ rr k [(1 − α 2 ψ f ( r , z ; r k , z k ) = µ 0 I k 2 ) J 1 ( α ) − J 2 ( α )] , B f = . . . απ 2D interpolation of magnetic measurements Compute ( β P , Q c , s , k ) k =1: N by least-square fit to magnetic measurements ψ in the vacuum Ω 0 \ (Ω p ∪ Ω C i ) Evaluate ( g , h ) = ( ψ, 1 r ∂ n ψ ) on Γ B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 11 / 19

Inverse problem step 2. Identification of the current density State equation  − ∆ ∗ ψ = λ [ r ψ ) + R 0  A ( ¯ r B ( ¯ ψ )]1 Ω p ( ψ ) in Ω R 0  ψ = g on Γ Least square minimization J ( A , B ) = J 0 + J ǫ with � (1 ∂ψ ∂ n − h ) 2 ds J 0 = r Γ � 1 � 1 ( ∂ 2 A ( ∂ 2 B ψ 2 ) 2 d ¯ ψ 2 ) 2 d ¯ J ǫ = ǫ ψ + ǫ ψ ∂ ¯ ∂ ¯ 0 0 B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 12 / 19

Numerical method Finite element resolution  Find ψ ∈ H 1 with ψ = g on Γ such that    � � 1 λ [ r ψ ) + R 0  ∀ v ∈ H 1 A ( ¯ r B ( ¯ µ 0 r ∇ ψ ∇ vdx = ψ )] vdx  0 ,  R 0 Ω Ω p with A ( x ) = � B ( ψ ) = � i a i f i ( x ) , i b i f i ( x ) , u = ( a i , b i ) Fixed point K ψ = 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 ] B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 13 / 19

Numerical method Least-square minimization J ( u ) = � C ψ − h � 2 + u T Au d : Neumann data A : regularization terms Approximation J ( u ) = � C ψ − d � 2 + u T Au , with ψ = K − 1 [ Y ( ψ n ) u + g ] = � CK − 1 Y ( ψ n ) u + CK − 1 g − d � 2 + u T Au J ( u ) = � E n u − F � 2 + u T Au Normal equation. Inverse solver : ψ n → u ( E nT E n + A ) u = E nT F B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 14 / 19

Algorithm. EQUINOX One equilibrium reconstruction : Fixed-point iterations : ◮ Inverse solver : ψ n → u n +1 ◮ Direct solver : ( ψ n , u n +1 ) → ψ n +1 ◮ Stopping condition || ψ n +1 − ψ n || < ǫ || ψ n || A pulse in real-time : Quasi-static approach : ◮ first guess at time t = equilibrium at time t − δ t ◮ limited number of iterations Normal equation : ≈ 10 basis func. → small ≈ 20 × 20 linear system Tikhonov regularization parameters unchanged K = LU and K − 1 , toroidal harmonics precomputed Expensive operations : update products CK − 1 Y ( ψ ) B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 15 / 19

Numerical Results : Tore Supra and JET characteristics ToreSupra JET Finite element mesh Number of triangles 1382 2871 Number of nodes 722 1470 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 B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 16 / 19

Tore Supra - Magnetics and polarimetry B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 17 / 19

JET - Magnetics and polarimetry B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 18 / 19

Conclusion Algorithm for equilibrium reconstruction and identification of the current density in real-time. EQUINOX Possibility to use internal measurements (interferometry, polarimetry, MSE) Robust identification of the averaged current density profile Makes possible future real-time control of current density profile Ref : Blum, Boulbe and Faugeras. Reconstruction of the equilibrium of the plasma in a Tokamak and identification of the current density profile in real time, JCP 231 (2012) 960-980 . B. Faugeras (Universit´ e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 19 / 19

Tore Supra. Magnetics and polarimetry.

Jet 68694. Magnetics only.

Jet 68694. Magnetics and polarimetry.