Finite Volume for Fusion Simulations Elise Estibals, Hervé Guillard, Afeintou Sangam To cite this version: Elise Estibals, Hervé Guillard, Afeintou Sangam. Finite Volume for Fusion Simulations. Jorek Meeting 2016, Apr 2016, Sophia Antipolis, France. <hal-01397086> HAL Id: hal-01397086 https://hal.inria.fr/hal-01397086 Submitted on 18 Nov 2016 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Finite Volume for Fusion Simulations E. Estibals H. Guillard A. Sangam elise.estibals@inria.fr Inria Sophia Antipolis Méditerranée November 15, 2016 1 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Plan MHD equations and issues 1 MHD equations and issues 2 Well-known methods Vector potential Divergence cleaning methods Proposed method 3 2-D system Evolution step 4 2 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Resistive MHD equations Non-conservative equations: E hd = E = B u + ηj p γ 1 + 1 2 ρu2, t ρ + (ρu) = 0, t (ρu) + (ρu u) + p = J B, t E hd + [(E hd + p)u ] = (J B) u + ηj 2, t B + E = 0. Conservative equations: E = E hd + 1 2 B2, p = p + 1 2 B2 t ρ + (ρu) = 0, t (ρu) + (ρu u B B) + p = 0, t E + [(E + p )u (u B)B] = (ηj B), t B + (u B B u) = (ηj). 3 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
MHD system MHD equations and issues Hyperbolic part of the system: 3 wave types 2 Alfvén waves, 4 magneto-acoustic waves (2 fast and 2 slow), 2 material waves (moving with u). Involution equation: Then Numerical issues: t ( B) = 0. B(t = 0) = 0 B(t) = 0 t. 1 Shock capturing: need a robust and accurate scheme. 2 constraint. 3 Time scale T >> T alfven = L/v alfven : need implicit scheme. 4 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Well-known methods Vector potential Divergence cleaning methods Proposed method constraint: Well-known methods 2 families of methods: Vector potential A: B = A. Divergence cleaning methods: Enforce B = 0. 5 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Vector potential A MHD equations and issues Well-known methods Vector potential Divergence cleaning methods Proposed method Replace B by A in the equation: t A u ( A) + η ( A) = U. Advantage: Insure B = 0. Drawbacks: One order higher in the spatial derivatives. More complex system of equation. 6 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Well-known methods Vector potential Divergence cleaning methods Proposed method Divergence cleaning methods: Previous works Powell et al. (JCP, 1999): Add a source term proportionnal to B. t B + (u B B u) = ( B)u. Munz et al. (JCP, 2000), Dedner et al. (JCP, 2001): Generalized Lagrange Multiplier (GLM) method. t B + (u B B u) + Ψ = 0, D(Ψ) + B = 0, D(Ψ) = 1 ch 2 t Ψ + 1 Ψ. cp 2 Balsara and Spicer (JCP, 1998): Constrained transport method. d B ds = E dl. dt 7 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation S S
Divergence cleaning methods Well-known methods Vector potential Divergence cleaning methods Proposed method Advantages: Easily incorporated with a Riemann type scheme. Drawbacks: Divergence cleaning: B appears in several equations. Not approximated with the same discretization. Divergence cleaning method insure B = 0 for one discrete approximation. Constrained transport: difficult to implement for arbitrary meshes. 8 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Proposed method MHD equations and issues Well-known methods Vector potential Divergence cleaning methods Proposed method A mixture of the two methods. Idea: work with a redundant system. Both vector potential and magnetic fields equations. Based on the relaxation scheme method: 1 Evolution step: FV method for all the variables. 2 Projection step: enforce B = A. 9 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
2-D implementation MHD equations and issues 2-D system Evolution step Scalar potential ψ: B = B z e z + e z ψ. Augmented system: t ρ + (ρu) = 0, t (ρu) + (ρu u B B) + p = 0, t E + [(E + p )u (u B)B] = η (J B), t B + (u B B u) = η 2 B, t ψ + u ψ = η 2 ψ. 10 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Evolution step MHD equations and issues 2-D system Evolution step System t U + x F (U) + y G(U) = 0: U n+1 i,j = U n i,j t x (F i+1/2,j F i 1/2,j ) t y (G i,j+1/2 G i,j 1/2 ). F i+1/2,j and G i,j+1/2 computed with a Riemann solver. 11 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Evolution step: HLLD scheme 2-D system Evolution step Miyoshi and Kusano (JCP, 2005). Approximate Riemann solver with 4 intermediate states. S L S L S M S R S R U L U L U R U R U L U R Figure : Riemann fan with four intermediate states. 12 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Evolution step: HLLD scheme 2-D system Evolution step Intermediate fluxes: { F K = F K + S K UK S K U K FK = F K + SK U K (S K S K )UK S, K = L, R. K U K ρψ flux: { ψl, 0 S F (ρψ) = F (ρ) M. ψ R, 0 S M 13 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
2-D system Evolution step 1 Evolution step: HLLD scheme { B n+1/2 i,j, ψ n+1/2 i,j. 2 Projection step: { ψ n+1 i,j = ψ n+1/2 i,j, B n+1 i,j = B n+1/2 z,i,j e z + e z ( ψ) n+1 i,j. 14 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
MHD equations and issues Shock capturing tests:,, Kelvin-Helmholtz. Plasma fusion tests:. 15 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
1-D shock tube Magnetic field: B = B x e x + x ψe y. 1-D equation for the potential ψ: Initial conditions: t (ρψ) + x (ρψu) = B x ρv. ρ u p B x B y = x ψ ψ x < 0.5 1 0 1 0.75 1 x x > 0.5 0.125 0 0.1 0.75 1 1 x 1-D mesh: 100 points Final time: 0.1 16 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
1-D : Magnetic component B y Figure : HLLD O(1). Figure : HLLD O(2). 17 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
2-D shock tube Potential: ψ(x, y) = 2-D mesh: 100 10 cells. { x 0.75y, x < 0.5 1 x 0.75y, x > 0.5. 18 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
2-D shock tube Figure : B y HLLD O(2). Figure : ψ, HLLD O(2). 19 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
MHD equations and issues Initial conditions: γ = 5/3 ρ u(x, y) v(x, y) p(x, y) B x (x, y) B y (x, y) ψ(x, y) γ 2 sin(2πy) sin(2πx) γ sin(2πy) sin(4πx) 1 2π cos(2πy) 1 4π cos(4πx) Mesh: 512 512 cells. Final time: 0.5. 20 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
: Pressure field Figure : Without projection. Figure : With projection. 21 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
MHD equations and issues Figure : Pressure at y = 0.3125 Red: without projection, Blue: with projection. 22 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
MHD equations and issues Figure : B. 23 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Initial data ρ p u(x, y) v(x, y) B x (x, y) B y (x, y) B z (x, y) ψ(x, y) 1 1 γ 1 2 tanh( y y 0 ) 0 0.1 cos( π 3 ) ρ 0 0.1 sin( π 3 ) ρ 0.1 cos( π 3 ) ρy Single mode pertubation v(x, y) = 0.01 sin(2πx) exp( y 2 Mesh: 256 512 points B pol B 2 B tor = x +By 2 B z. ), σ = 0.01. σ2 24 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Kelvin-Helmholtz: t = 5.0 Figure : Without projection. Figure : With projection. 25 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Kelvin-Helmholtz: t = 8.0 Figure : Without projection. Figure : With projection. 26 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Kelvin-Helmholtz: t = 12.0 Figure : Without projection. Figure : With projection. 27 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Kelvin-Helmholtz: t = 20.0 Figure : Without projection. Figure : With projection. 28 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Kelvin-Helmholtz MHD equations and issues Figure : B. 29 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Preliminary test for fusion plasma Equilibrium computation: p = J B. Initial conditions: R 0 = 10, B r = 0, r ρ = 1, B θ (r) = R 0 (3r 2 +1), u = 0, B z = 1, 1 p(r) = 6R0 2(3r, ψ(r) = 1 2 +1) 2 6R 0 ln(3r 2 + 1). Aligned mesh: cylindrical coordinates (100 10 cells). Non aligned mesh: Cartesian coordinates (200 200 cells). 30 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Cylindrical coordinates Figure : B θ. Figure : p. 31 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Cylindrical coordinates: steady state Figure : Residual on r-momentum equation. Figure : Relative error of B θ. 32 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Cartesian coordinates Figure : Relative error of p in function of Alfvén time. Non aligned meshes: with projection simulations maintains the equilibrium on 1000 Alfvén times (around 0.5ms and 10 6 time steps). However, we want to perform the simulation on several milliseconds. 33 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Remedies MHD equations and issues A simple trick: remove the truncation error evaluated at time t = 0. t (ρu) + (ρu u) + p J B ( h p J h B h ) eq = 0. Well-balanced scheme: work in progress. Figure : Relative error of p. 34 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Conclusions and perspectives Conclusions: Shock capturing tests: gives satisfactory results. Plasma fusion test: Work in progress on well-balanced scheme. Perspectives: Perform more tests for plasma fusion (kink instability). Adapt to 3-D geometry. Test on resistive problems. 35 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
Cylindrical equations MHD equations and issues t (rρ) + r (rρu [ r ) + θ (ρu θ ) ] = 0, t (rρu r ) + r r(ρur 2 + p Br 2 ) + θ (ρu r u θ B r B θ ) = ρuθ 2 + p B 2 ] ] θ, t (r 2 ρu θ ) + r [r 2 (ρu r u θ B r B θ ) + θ [r(ρuθ 2 + p Bθ 2 ) = 0, t (rρu z ) + [ r [r(ρu r u z B r B z )] + θ (ρu θ u z B θ B z ) = 0, t (re) + r r[(e + p )ur (u B)B r ] ] [ + θ (E + p ] )uθ (u B)B θ = 0, t (rb r ) + θ (u θ B r u r B θ ) = 0, t B θ + r (u r B θ u θ B r ) = 0, t (rb z ) + r [r(u r B z u z B r )] + θ (u θ B z u z B θ ) = 0, t (rρψ) + r (rρψu r ) + θ (ρψu θ ) = 0. 36 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation
1 Evolution step: HLLD scheme 2 Projection step: p n+1/2 i,j, B n+1/2 i,j, ψ n+1/2 i,j. ψ n+1 i,j = ψ n+1/2 i,j, B n+1 i,j = B n+1/2 z,i,j e z + e z ( ψ) n+1 i,j, E n+1 i,j = pn+1/2 i,j γ 1 + 1 2 ρn+1/2 u 2 + 1 2 (Bn+1 i,j ) 2. 37 / 35 E. Estibals, H. Guillard, A. Sangam FV method for fusion simulation