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 = µj (Ampere’s law)

Axisymmetric assumption => Grad-Shafranov equation

2D problem. Cylindrical coordinates (r, φ, z) State variable ψ(r, z) poloidal magnetic flux Bp = 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 µ0r ∂ψ ∂r ) + ∂ ∂z ( 1 µ0r ∂ψ ∂z ) = rp′(ψ) + 1 µ0r (ff ′)(ψ)

In the vacuum

−∆∗ψ = 0

Boundary value problem

   −∆∗ψ = [rp′(ψ) + 1 µ0r ff ′(ψ)]1Ωp(ψ) in Ω ψ = g

• n

Γ

• B. Faugeras (Universit´

e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 6 / 19

Definition of the free plasma boundary

Two cases

• utermost 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

Computational domain

z r ΩCi Ω0

B probe

Ωp

limiter

Ω

⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

Γ flux loop

b

C −∆∗ψ = 0 −∆∗ψ = j(r, ψ) −∆∗ψ = ji

• 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(Mi) B probes : Bp(Ni).di

Cauchy conditions (ψ, ∂nψ) 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 2π]

Quasi-separable solutions in bipolar (toroidal) coordinates

• Ψ(τ, η, φ) =
• cosh τ − cosh ηA(τ)B(η) cos φ

Complete set of solutions

• T P,Q

c,s,k

• k∈N =

a sinh τ √cosh τ − cos η P1

k− 1

2 (cosh τ)

Q1

k− 1

2 (cosh τ)

• cos(kη)

sin(kη)

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

ψ(r, z) =

N

• k=0

(βP,Q

c,s,k)(T P,Q c,s,k) +

• k

ψf (r, z; rk, zk) B(r, z) =

N

• k=0

(βP,Q

c,s,k)Bk(r, z) +

• k

Bf (r, z; rk, zk)

PF coils modelized by filaments of current

Current Ik at (rk, zk) : ψf (r, z; rk, zk) = µ0Ik απ √rrk[(1 − α2 2 )J1(α) − J2(α)], Bf = . . .

2D interpolation of magnetic measurements

Compute (βP,Q

c,s,k)k=1:N by least-square fit to magnetic measurements

ψ in the vacuum Ω0 \ (Ωp ∪ ΩCi) Evaluate (g, h) = (ψ, 1

r ∂nψ) 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 R0 A( ¯ ψ) + R0 r B( ¯ ψ)]1Ωp(ψ) in Ω ψ = g

• n Γ

Least square minimization

J(A, B) = J0 + Jǫ with J0 =

• Γ

(1 r ∂ψ ∂n − h)2ds Jǫ = ǫ 1 (∂2A ∂ ¯ ψ2 )2d ¯ ψ + ǫ 1 (∂2B ∂ ¯ ψ2 )2d ¯ ψ

• B. Faugeras (Universit´

e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 12 / 19

Numerical method

Finite element resolution

       Find ψ ∈ H1 with ψ = g on Γ such that ∀v ∈ H1

0,

• Ω

1 µ0r ∇ψ∇vdx =

• Ωp

λ[ r R0 A( ¯ ψ) + R0 r B( ¯ ψ)]vdx with A(x) =

i aifi(x),

B(ψ) =

i bifi(x), u = (ai, bi)

Fixed point

Kψ = Y (ψ)u + g K modified stiffness matrix, u coefficients of A and B, g Dirichlet BC

Direct solver : (ψn, u) → ψn+1

ψn+1 = K −1[Y (ψn)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) = Cψ − h2 + uTAu d : Neumann data A : regularization terms

Approximation

J(u) = Cψ − d2 + uTAu, with ψ = K −1[Y (ψn)u + g] J(u) = CK −1Y (ψn)u + CK −1g − d2 + uTAu = E nu − F2 + uTAu

Normal equation. Inverse solver : ψn → u

(E nTE n + A)u = E nTF

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e de Nice) Equilibrium reconstruction in a Tokamak PICOF, April 2012 14 / 19

• Algorithm. EQUINOX

One equilibrium reconstruction :

Fixed-point iterations :

◮ Inverse solver : ψn → un+1 ◮ Direct solver : (ψn, un+1) → ψn+1 ◮ Stopping condition ||ψn+1 − ψn||

||ψn|| < ǫ

A pulse in real-time :

Quasi-static approach :

◮ first guess at time t = equilibrium at time t − δt ◮ 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 −1Y (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.