Finite Elements for Magnetohydrodynamics and its Optimal Control

Transcription

1 Finite Elements for Magnetohydrodynamics and its Karl Kunisch Marco Discacciati (RICAM) FEM Symposium Chemnitz September 25 27, 2006

2 Overview 1 2 3

3 What is Magnetohydrodynamics? Magnetohydrodynamics (MHD) concerns the mutual interaction of electrically conducting fluids and magnetic fields

4 What is Magnetohydrodynamics? Magnetohydrodynamics (MHD) concerns the mutual interaction of electrically conducting fluids and magnetic fields Use taylored magnetic fields for... stirring of conducting fluids flow damping (during casting or solidification) electromagnetic filtration, melting, levitation

5 Application: Casting of Aluminum Illustration: B.Q. Li

6 Application: Casting of Aluminum Features convection flow due to temperature gradients undesired inflow of impurities idea: damping by magnetic fields Illustration: B.Q. Li

7 Application: Production of Aluminum Solidified bath Anods Electrolytic bath Interface Liquid aluminum Cathod Illustration: J.-F. Gerbeau

8 Application: Production of Aluminum Solidified bath Anods Electrolytic bath Interface Liquid aluminum Cathod Features two fluids, free surface, free interface electrolytic bath shallow: huge energy savings electrolytic bath deep: damping of instabilities (stray magnetic fields) Illustration: J.-F. Gerbeau

9 Application: CZ Crystal Growth Illustration: B.Q. Li

10 Application: CZ Crystal Growth Features convection-driven flow free surface, Marangoni effect, non-local radiation idea: damping or stirring by magnetic fields Illustration: B.Q. Li

11 Application: FZ Crystal Growth Illustration: B.Q. Li

12 Application: FZ Crystal Growth Features two free interfaces free surface no mechanical support for melt phase idea: heat, confine and shape the melt phase by magnetic fields Illustration: B.Q. Li

13 Summary: Applications of MHD Numerous applications in... metallurgy crystal growth

14 Summary: Applications of MHD Numerous applications in... metallurgy crystal growth Most attractive features contactless application of a volume force (Lorentz force) induction heating

15 Interaction in MHD Interaction principles charge carries moving in magnetic field induce currents

16 Interaction in MHD Interaction principles charge carries moving in magnetic field induce currents currents induce magnetic fields

17 Interaction in MHD Interaction principles charge carries moving in magnetic field induce currents currents induce magnetic fields magnetic fields exert a Lorentz force on moving charge carries

18 MHD Equations: The Stationary Case Navier-Stokes system with Lorentz force ϱ (u )u η u + p = J B u = 0 fluid velocity u on Ω pressure p on Ω current density J on Ω magnetic field B on R 3

19 MHD Equations: The Stationary Case Navier-Stokes system with Lorentz force ϱ (u )u η u + p = J B u = 0 fluid velocity u on Ω pressure p on Ω Charge conservation and Ohm s law σ 1 J + φ = u B J = 0 current density J on Ω electric potential φ on Ω magnetic field B on R 3

20 MHD Equations: The Stationary Case Navier-Stokes system with Lorentz force ϱ (u )u η u + p = J B u = 0 fluid velocity u on Ω pressure p on Ω Charge conservation and Ohm s law σ 1 J + φ = u B J = 0 current density J on Ω electric potential φ on Ω No monopoles and Ampère s law B = 0 and (µ 1 B) = J magnetic field B on R 3

21 MHD Equations: The Stationary Case Navier-Stokes system with Lorentz force ϱ (u )u η u + p = J B u = 0 fluid velocity u on Ω pressure p on Ω Charge conservation and Ohm s law σ 1 J + φ = u B J = 0 current density J on Ω electric potential φ on Ω No monopoles and Ampère s law B = 0 and (µ 1 B) = J magnetic field B on R 3

22 MHD Equations: The Stationary Case Navier-Stokes system with Lorentz force ϱ (u )u η u + p = J B u = 0 u = h Charge conservation and Ohm s law σ 1 J + φ = u B J = 0 J n = j φ = φ c No monopoles and Ampère s law B = 0 and (µ 1 B) = J

23 MHD Equations: The Stationary Case Navier-Stokes system with Lorentz force ϱ (u )u η u + p = J B u = 0 Charge conservation and Ohm s law σ 1 J + φ = u B J = 0 u = h J n = j φ = φ c Elimination of magnetic field velocity current formulation B = B(J)(x) = µ x y J(y) dy Biot Savart law 4π x y 3 R 3

24 MHD Equations: Velocity Current Formulation Coupled system ϱ (u )u η u + p = J B(J) u = 0 σ 1 J + φ = u B(J) J = 0 u = h J n = j φ = φ c

25 MHD Equations: Velocity Current Formulation Coupled system ϱ (u )u η u + p = J ( ) B(J) + B 0 u = h u = 0 σ 1 J + φ = u ( ) B(J) + B 0 J n = j J = 0 φ = φ c

26 MHD Equations: Velocity Current Formulation Coupled system Note ϱ (u )u η u + p = J ( B(J) + B 0 ) u = 0 σ 1 J + φ = u ( B(J) + B 0 ) J = 0 all quantities confined to fluid domain Ω u = h J n = j φ = φ c

27 MHD Equations: Velocity Current Formulation Coupled system Note ϱ (u )u η u + p = J ( B(J) + B 0 ) u = 0 σ 1 J + φ = u ( B(J) + B 0 ) J = 0 all quantities confined to fluid domain Ω some adjustable quantities, e.g., φ c u = h J n = j φ = φ c

28 MHD Equations: Velocity Current Formulation Coupled system Note ϱ (u )u η u + p = J ( B(J) + B 0 ) u = 0 σ 1 J + φ = u ( B(J) + B 0 ) J = 0 all quantities confined to fluid domain Ω some adjustable quantities, e.g., φ c saddle point structure u = h J n = j φ = φ c

29 MHD Equations: Analysis Nonlinear saddle point problem A 0 (u, J) + A 1 ((u, J), (u, J)) + B (p, φ) = F B(u, J) = 0

30 MHD Equations: Analysis Nonlinear saddle point problem A 0 (u, J) + A 1 ((u, J), (u, J)) + B (p, φ) = F B(u, J) = 0 Solution u H 1 (Ω) J H(div; Ω) = {J L 2 (Ω): J L 2 (Ω)} B(J) H 1 (R 3 ) p L 2 (Ω)/R φ L 2 (Ω)/R [1]; Meir, Schmidt: SIAM Journal on Numerical Analysis, 1999 [2]: Griesse, Kunisch: SIAM Journal on Control and Optimization, to appear

31 Related Work Previous and ongoing work M. Gunzburger, A.J. Meir, P. Schmidt J.-F. Gerbeau, C. Le Bris, T. Lelièvre J. Rappaz, R. Touzani many authors in engineering S. Hou, J. Peterson, A.J. Meir, S.S. Ravindran M. Hinze and co-workers M. Gunzburger, C. Trenchea

32 Overview 1 2 3

33 FEM Discretization Conforming and stable discretization (u, p) H 1 (Ω) L 2 (Ω)/R

34 FEM Discretization Conforming and stable discretization (u, p) H 1 (Ω) L 2 (Ω)/R Taylor-Hood elements

35 FEM Discretization Conforming and stable discretization (u, p) H 1 (Ω) L 2 (Ω)/R Taylor-Hood elements (J, φ) L 2 (div; Ω) L 2 (Ω)/R

36 FEM Discretization Conforming and stable discretization (u, p) H 1 (Ω) L 2 (Ω)/R Taylor-Hood elements (J, φ) L 2 (div; Ω) L 2 (Ω)/R Raviart-Thomas elements

37 FEM Discretization Conforming and stable discretization (u, p) H 1 (Ω) L 2 (Ω)/R Taylor-Hood elements (J, φ) L 2 (div; Ω) L 2 (Ω)/R Raviart-Thomas elements Discrete stability condition inf sup (q,ψ) (u,j) b((u, J), (q, ψ)) (u, J) (q, ψ) β

38 FEM Discretization Biot-Savart law If J = 0... B(J)(x) = µ 4π R 3 x y J(y) dy x y 3 B = µj on R 3 B = 0 on R 3

39 FEM Discretization Biot-Savart law B(J)(x) = µ 4π Introduction of vector potential B = A R 3 x y J(y) dy x y 3 B = µj on R 3 B = 0 on R 3 ( A) = µj on R 3

40 FEM Discretization Biot-Savart law B(J)(x) = µ 4π Introduction of vector potential B = A R 3 x y J(y) dy x y 3 B = µj on R 3 B = 0 on R 3 ( A) = µj on R 3 A = 0 on R 3 (gauging)

41 FEM Discretization Biot-Savart law B(J)(x) = µ 4π Introduction of vector potential B = A R 3 x y J(y) dy x y 3 ( A) = µj on R 3 A = 0 on R 3 (gauging)

42 FEM Discretization Biot-Savart law B(J)(x) = µ 4π Introduction of vector potential B = A R 3 x y J(y) dy x y 3 ( A) = µj on Ω A A = 0 on Ω A (gauging)

43 FEM Discretization Biot-Savart law B(J)(x) = µ 4π Introduction of vector potential B = A R 3 x y J(y) dy x y 3 ( A) = µj on Ω A A = 0 on Ω A (gauging) A n = 0 on Γ A

44 FEM Discretization Biot-Savart law B(J)(x) = µ 4π Introduction of vector potential B = A R 3 x y J(y) dy x y 3 ( A) = µj on Ω A A = 0 on Ω A (gauging) A n = 0 on Γ A Solvability condition J = 0 on Ω A

45 FEM Discretization Curl curl equation ( A) = µj on Ω A A = 0 on Ω A A n = 0 on Γ A

46 FEM Discretization Curl curl equation ( A) = µj on Ω A A = 0 on Ω A A n = 0 on Γ A Existence and uniqueness For J = 0, there exists a unique solution in H(curl; Ω A ).

47 FEM Discretization Curl curl equation ( A) = µj on Ω A A = 0 on Ω A A n = 0 on Γ A Existence and uniqueness For J = 0, there exists a unique solution in H(curl; Ω A ). Conforming discretization A H(curl; Ω A )

48 FEM Discretization Curl curl equation ( A) = µj on Ω A A = 0 on Ω A A n = 0 on Γ A Existence and uniqueness For J = 0, there exists a unique solution in H(curl; Ω A ). Conforming discretization A H(curl; Ω A ) Nédélec elements

49 FEM Discretization Condition on the boundary conditions J = 0 on Ω A

50 FEM Discretization Condition on the boundary conditions J = 0 on Ω A { J = 0 on Ω J n = 0 on Ω

51 FEM Discretization Condition on the boundary conditions J = 0 on Ω A { J = 0 on Ω J n = 0 on Ω Unless Ω = Ω A, this excludes φ = φ c!

52 FEM Discretization Condition on the boundary conditions J = 0 on Ω A { J = 0 on Ω J n = 0 on Ω Unless Ω = Ω A, this excludes φ = φ c! Use [φ] Γφ = φ c instead.

53 FEM Discretization Condition on the boundary conditions J = 0 on Ω A { J = 0 on Ω J n = 0 on Ω Unless Ω = Ω A, this excludes φ = φ c! Use [φ] Γφ = φ c instead. [Hiptmair, Sterz]

54 FEM Discretization Newton system M F δa G[J] A[u] B C[A] δu B δp H[u] C[A] D E δj = b E δφ

55 FEM Discretization Newton system M F δa G[J] A[u] B C[A] δu B δp H[u] C[A] D E δj = b E δφ

56 FEM Discretization Newton system M F δa G[J] A[u] B C[A] δu B δp H[u] C[A] D E δj = b E δφ

57 FEM Discretization Newton system M F δa G[J] A[u] B C[A] δu B δp H[u] C[A] D E δj = b E δφ

58 FEM Discretization Newton system M F δa G[J] A[u] B C[A] δu B δp H[u] C[A] D E δj = b E δφ Challenges mixture of finite element spaces preconditioning of linear systems

59 Simulation Results (joint with M. Discacciati) 9.506e e e e e 04 Problem description B 0 induced by currents in wires

60 Simulation Results (joint with M. Discacciati) 9.506e e e e e 04 Problem description B 0 induced by currents in wires current J n = ±j at top/bottom (Ω = Ω A )

61 Simulation Results (joint with M. Discacciati) Fluid velocity (from top, slice at half height), Stokes 3.690e e e e e 03 Problem description B 0 induced by currents in wires current J n = ±j at top/bottom (Ω = Ω A ) two counter-rotating flow cells ([Gerbeau], 2000)

62 Simulation Results Problem size order of Nédélec FE Taylor-Hood FE Raviart-Thomas FE Grid 1 dofs (344 tetr.) iterations Grid 2 dofs (2752 tetr.) iterations Details iterative damped splitting scheme: A and (u, p, J, φ) implementation in Ngsolve, sparse direct solver Pardiso

63 Overview 1 2 3

64 Problem formulation Minimize f (y, u) subject to e(y, u) = 0

65 Problem formulation Minimize f (y, u) subject to e(y, u) = 0 Lagrange functional L(y, u, λ) := f (y, u) + e(y, u), λ λ adjoint state

66 Problem formulation Minimize f (y, u) subject to e(y, u) = 0 Lagrange functional L(y, u, λ) := f (y, u) + e(y, u), λ λ adjoint state Necessary conditions

67 Problem formulation Minimize f (y, u) subject to e(y, u) = 0 Lagrange functional L(y, u, λ) := f (y, u) + e(y, u), λ λ adjoint state Necessary conditions L y (y, u, λ) = f y (y, u) + e y (y, u) λ = 0

68 Problem formulation Minimize f (y, u) subject to e(y, u) = 0 Lagrange functional L(y, u, λ) := f (y, u) + e(y, u), λ λ adjoint state Necessary conditions L y (y, u, λ) = f y (y, u) + e y (y, u) λ = 0 adjoint PDE system

69 Problem formulation Minimize f (y, u) subject to e(y, u) = 0 Lagrange functional L(y, u, λ) := f (y, u) + e(y, u), λ λ adjoint state Necessary conditions L y (y, u, λ) = f y (y, u) + e y (y, u) λ = 0 L u (y, u, λ) = f u (y, u) + e u(y, u) λ = 0 adjoint PDE system

70 Problem formulation Minimize f (y, u) subject to e(y, u) = 0 Lagrange functional L(y, u, λ) := f (y, u) + e(y, u), λ λ adjoint state Necessary conditions L y (y, u, λ) = f y (y, u) + e y (y, u) λ = 0 L u (y, u, λ) = f u (y, u) + e u(y, u) λ = 0 adjoint PDE system gradient equation

71 Problem formulation Minimize f (y, u) subject to e(y, u) = 0 Lagrange functional L(y, u, λ) := f (y, u) + e(y, u), λ λ adjoint state Necessary conditions L y (y, u, λ) = f y (y, u) + e y (y, u) λ = 0 L u (y, u, λ) = f u (y, u) + e u(y, u) λ = 0 L λ (y, u, λ) = e(y, u) = 0 adjoint PDE system gradient equation

72 Problem formulation Minimize f (y, u) subject to e(y, u) = 0 Lagrange functional L(y, u, λ) := f (y, u) + e(y, u), λ λ adjoint state Necessary conditions L y (y, u, λ) = f y (y, u) + e y (y, u) λ = 0 L u (y, u, λ) = f u (y, u) + e u(y, u) λ = 0 L λ (y, u, λ) = e(y, u) = 0 adjoint PDE system gradient equation PDE system

73 Newton s method in function space L yy L yu e y δy L y L uy L uu eu δu = L u e y e u 0 δλ e }{{} KKT matrix

74 Newton s method in function space L yy L yu e y δy L y L uy L uu eu δu = L u e y e u 0 δλ e }{{} KKT matrix Numerical challenges

75 Newton s method in function space L yy L yu e y δy L y L uy L uu eu δu = L u e y e u 0 δλ e }{{} KKT matrix Numerical challenges large system of equations ( variables)

76 Newton s method in function space L yy L yu e y δy L y L uy L uu eu δu = L u e y e u 0 δλ e }{{} KKT matrix Numerical challenges large system of equations ( variables) symmetric, indefinite, ill conditioned

77 An MHD Problem A possible problem setup

78 An MHD Problem A possible problem setup fluid region Ω

79 An MHD Problem A possible problem setup electrodes fluid region Ω

80 An MHD Problem A possible problem setup electrodes fluid region Ω Purpose and control mechanisms influence flow pattern (stir or dampen)

81 An MHD Problem A possible problem setup electrodes fluid region Ω Purpose and control mechanisms influence flow pattern (stir or dampen) using adjustable quantities

82 An MHD Problem A possible problem setup electrodes φ = { φ c control R 0 fluid region Ω Purpose and control mechanisms influence flow pattern (stir or dampen) using adjustable quantities (applied potential difference)

83 An MHD Problem A possible problem setup electrodes φ = { φ c control R 0 fluid region Ω = Ω A Purpose and control mechanisms influence flow pattern (stir or dampen) using adjustable quantities (applied potential difference)

84 An MHD Problem Problem description Minimize J = α 2 u u d 2 L 2 (Ω) + γ 2 φ c 2 s.t. MHD system

85 An MHD Problem Problem description Minimize J = α 2 u u d 2 L 2 (Ω) + γ 2 φ c 2 s.t. MHD system φ = φ c at electrode 1 φ = 0 at electrode 2 J n = 0 elsewhere

86 An MHD Problem Problem description Minimize J = α 2 u u d 2 L 2 (Ω) + γ 2 φ c 2 s.t. MHD system φ = φ c at electrode 1 φ = 0 at electrode 2 J n = 0 elsewhere Given problem data u d desired velocity field; cost parameters α 0 and γ 0

87 An MHD Problem Problem description Minimize J = α 2 u u d 2 L 2 (Ω) + γ 2 φ c 2 s.t. MHD system φ = φ c at electrode 1 φ = 0 at electrode 2 J n = 0 elsewhere Given problem data u d desired velocity field; cost parameters α 0 and γ 0 applied magnetic field B 0

88 An MHD Problem Problem description Given problem data Minimize J = α 2 u u d 2 L 2 (Ω) + γ 2 φ c 2 s.t. MHD system φ = φ c at electrode 1 φ = 0 at electrode 2 J n = 0 elsewhere u d desired velocity field; cost parameters α 0 and γ 0 applied magnetic field B 0 u = h on the boundary Ω

89 An MHD Problem Adjoint system on Ω ϱ ( u) v ϱ (u ) v η v + q (B(J) + B 0 ) K = σ 1 K B(K u + v J) + ψ (B(J) + B 0 ) v = Incompressibility and boundary conditions v = 0 on Ω v = 0 on Ω K = 0 on Ω K n = 0 or ψ = 0 on Ω Adjoint variables on Ω v adjoint velocity K adjoint current q adjoint pressure ψ adjoint potential

90 An MHD Problem Adjoint system on Ω ϱ ( u) v ϱ (u ) v η v + q (B(J) + B 0 ) K = σ 1 K B(K u + v J) + ψ (B(J) + B 0 ) v = Incompressibility and boundary conditions v = 0 on Ω v = 0 on Ω K = 0 on Ω K n = 0 or ψ = 0 on Ω Adjoint variables on Ω v adjoint velocity K adjoint current q adjoint pressure ψ adjoint potential

91 An MHD Problem Adjoint system on Ω ϱ ( u) v ϱ (u ) v η v + q (B(J) + B 0 ) K = σ 1 K B(K u + v J) + ψ (B(J) + B 0 ) v = Incompressibility and boundary conditions v = 0 on Ω v = 0 on Ω K = 0 on Ω K n = 0 or ψ = 0 on Ω Adjoint variables on Ω v adjoint velocity K adjoint current q adjoint pressure ψ adjoint potential

92 An MHD Problem Adjoint system on Ω ϱ ( u) v ϱ (u ) v η v + q (B(J) + B 0 ) K = σ 1 K B(K u + v J) + ψ (B(J) + B 0 ) v = Incompressibility and boundary conditions v = 0 on Ω v = 0 on Ω K = 0 on Ω K n = 0 or ψ = 0 on Ω Optimality condition γφ c + K n = 0 at electrodes

93 Numerics for KKT matrix G[λ J ] H[λ u ] M G[J] H[u] G[λ J ] A[u] B C[A] B H[λ u ] F C[A] D E E γ M F G[J] A[u] B C[A] B H[u] C[A] D E E

94 Numerics for KKT matrix G[λ J ] H[λ u ] M G[J] H[u] G[λ J ] A[u] B C[A] B H[λ u ] F C[A] D E E γ M F G[J] A[u] B C[A] B H[u] C[A] D E E

95 Numerics for KKT matrix G[λ J ] H[λ u ] M G[J] H[u] G[λ J ] A[u] B C[A] B H[λ u ] F C[A] D E E γ M F G[J] A[u] B C[A] B H[u] C[A] D E E

96 Numerics for KKT matrix Simplifications: G[λ J ] H[λ u ] M G[J] H[u] G[λ J ] A[u] B C[A] B H[λ u ] F C[A] D E E γ M F G[J] A[u] B C[A] B H[u] C[A] D E E

97 Numerics for KKT matrix Simplifications: Stokes flow G[λ J ] H[λ u ] M G[J] H[u] G[λ J ] A[u] B C[A] B H[λ u ] F C[A] D E E γ M F G[J] A[u] B C[A] B H[u] C[A] D E E

98 Numerics for KKT matrix Simplifications: Stokes flow G[λ J ] H[λ u ] M G[J] H[u] G[λ J ] A B C[A] B H[λ u ] F C[A] D E E γ M F G[J] A B C[A] B H[u] C[A] D E E

99 Numerics for KKT matrix Simplifications: Stokes flow, B = B 0 known (low R m approx.) G[λ J ] H[λ u ] M G[J] H[u] G[λ J ] A B C[A] B H[λ u ] F C[A] D E E γ M F G[J] A B C[A] B H[u] C[A] D E E

100 Numerics for KKT matrix Simplifications: Stokes flow, B = B 0 known (low R m approx.) G[λ J ] H[λ u ] M G[J] H[u] G[λ J ] A B C[A] B H[λ u ] F C[A] D E E γ M F G[J] A B C[A] B H[u] C[A] D E E

101 Numerical Results (joint with M. Discacciati) Problem data

102 Numerical Results (joint with M. Discacciati) Problem data grounded electrode (φ = 0)

103 Numerical Results (joint with M. Discacciati) Problem data control electrode (φ = φ c ) grounded electrode (φ = 0)

104 Numerical Results (joint with M. Discacciati) Problem data control electrode (φ = φ c ) grounded electrode (φ = 0) B 0 = 10 4 (0, 0, x)t

105 Numerical Results (joint with M. Discacciati) Problem data control electrode (φ = φ c ) grounded electrode (φ = 0) B 0 = 10 4 (0, 0, x)t u d = swirl flow

106 Numerical Results (joint with M. Discacciati) Problem data control electrode (φ = φ c ) grounded electrode (φ = 0) B 0 = 10 4 (0, 0, x)t u d = swirl flow Material data and solution material data of liquid Al at 700 C optimal control φ c = V at γ = 0.1 current J max = A/m 2, velocity u max = m/s R m = µσul =

107 Numerical Results Problem discretization 768 tetrahedra Taylor-Hood (u, p) order 2 1 Raviart-Th. (J, φ) order degrees of freedom implementation in Ngsolve CPU time matrix setup 30 s sparse factorization (Pardiso) 36 s solution of optimal control problem < 1 s

108 Numerical Results Optimal solution (potential φ)

109 Numerical Results Optimal solution (current J)

110 Numerical Results Optimal solution (velocity u)

111 Numerical Results Optimal solution (velocity u)

112 Concluding Remarks Summary simulation and optimization problems in MHD numerous applications in metallurgy, crystal growth discretization: Taylor-Hood, Raviart-Thomas and Nédélec FE

113 Concluding Remarks Summary simulation and optimization problems in MHD numerous applications in metallurgy, crystal growth discretization: Taylor-Hood, Raviart-Thomas and Nédélec FE Future challenges time-dependent and Navier-Stokes cases adaptivity preconditioning of linear systems

114 Concluding Remarks Summary simulation and optimization problems in MHD numerous applications in metallurgy, crystal growth discretization: Taylor-Hood, Raviart-Thomas and Nédélec FE Future challenges time-dependent and Navier-Stokes cases adaptivity preconditioning of linear systems Thank you!