Control of Interface Evolution in Multi-Phase Fluid Flows

Transcription

1 Control of Interface Evolution in Multi-Phase Fluid Flows Markus Klein Department of Mathematics University of Tübingen Workshop on Numerical Methods for Optimal Control and Inverse Problems Garching, Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 1 / 17

2 Contents Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 2 / 17

3 Contents 1 Introduction and Motivation 2 Analysis 3 Numerical Analysis and Computations Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 3 / 17

4 The Model ρ 0 = ρ 1 χ Ω1 + ρ 2 χ Ω2 mixture of two immiscible viscous incompressible fluids in a bounded domain in R 2. Multi-phase flow evolution by Navier Stokes Eq. (cf. [Lions, 1996]) ρy t + ρ[y ]y µ y + p = ρu, y(0) = y 0, (NSE) ρ t + [y ]ρ = 0, ρ(0) = ρ 0, div y = 0 + B.C. Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 4 / 17

5 The Model ρ 0 = ρ 1 χ Ω1 + ρ 2 χ Ω2 mixture of two immiscible viscous incompressible fluids in a bounded domain in R 2. Multi-phase flow evolution by Navier Stokes Eq. (cf. [Lions, 1996]) Minimize Shape Geometry Cost T J(ρ, u) = ρ(t) σ 2 dx dt + β T H 1 (S ρ ) dt + α T u 2 dx dt 0 Ω Ω subject to ρy t + ρ[y ]y µ y + p = ρu, y(0) = y 0, (NSE) ρ t + [y ]ρ = 0, ρ(0) = ρ 0, div y = 0 + B.C. Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 4 / 17

6 Evidence of the geometric functional ρ σ 2 L 2 (Ω T ) ρ σ 2 L 2 (Ω T ) Target σ + T 0 H 1 (S ρ ) Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 5 / 17

7 Evidence of the geometric functional ρ σ 2 L 2 (Ω T ) ρ σ 2 L 2 (Ω T ) Target σ + T 0 H 1 (S ρ ) Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 5 / 17

8 Evidence of the geometric functional ρ σ 2 L 2 (Ω T ) ρ σ 2 L 2 (Ω T ) Target σ + T 0 H 1 (S ρ ) better corners correct geometry Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 5 / 17

9 Application ([Gerbeau et al., 2006]): Aluminium production via electrolysis Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 6 / 17

10 Application ([Gerbeau et al., 2006]): Aluminium production via electrolysis Anods shall not touch the interface! Interface control Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 6 / 17

11 Goals Existence of optimum. (Necessary) first order optimality conditions. Numerical scheme with low order Finite Elements. Convergence of the numerical scheme. Known result Optimization (analysis, no numerics) of L 2 -functional (no geometric term) subject to Stokes equation, cf. [Kunisch and Lu, 2011]. Convergent numerical scheme for equation (low regularity), cf. [Ba nas and Prohl, 2010]. Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 7 / 17

12 Contents 1 Introduction and Motivation 2 Analysis 3 Numerical Analysis and Computations Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 8 / 17

13 Analytical problems and strategy Minimize T J(ρ, u) = ρ(t) σ 2 dx dt + β T H 1 (S ρ ) dt + α T u 2 dx dt 0 Ω Ω subject to ρy t + ρ[y ]y µ y + p = ρu, y(0) = y 0, (NSE) ρ t + [y ]ρ = 0, ρ(0) = ρ 0, div y = 0 + B.C. Problem: Not clear if red term is w.l.s.c., and not clear if corresponding Lagrange multiplier to mass equation exists and is a function. Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 9 / 17

14 Analytical problems and strategy Minimize T J(ρ, u) = ρ(t) σ 2 dx dt + β T H 1 (S ρ ) dt + α T u 2 dx dt 0 Ω Ω subject to ρy t + ρ[y ]y µ y + p = ρu, y(0) = y 0, (NSE) ρ t + [y ]ρ = 0, ρ(0) = ρ 0, div y = 0 + B.C. Problem: Not clear if red term is w.l.s.c., and not clear if corresponding Lagrange multiplier to mass equation exists and is a function. Solution: Add artificial diffusion to equation and approximate Hausdorff measure ( Mortola-Modica, cf. [Braides, 1998]) Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 9 / 17

15 Analytical problems and strategy Minimize J δ (ρ, u) = + β 2 ( δ ρ ) W (ρ) Ω T δ Ω T subject to ρy t + ρ[y ]y µ y + p = ρu, y(0) = y 0, (NSE ε ) ρ t + [y ]ρ ε ρ t = 0, ρ(0) = ρ 0, div y = 0 + B.C. (W 0 double Well functional with W (ρ) = 0 iff ρ = ρ 1 or ρ = ρ 2 ) + Solution: Add artificial diffusion to equation and approximate Hausdorff measure ( Mortola-Modica, cf. [Braides, 1998]) Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 9 / 17

16 Analytic results Theorem (Existence) For δ, ε > 0, there exists at least one minimum and the corresponding Lagrange multipliers belong to some L p (Ω T ) for p > 1. Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 10 / 17

17 Analytic results Theorem (Existence) For δ, ε > 0, there exists at least one minimum and the corresponding Lagrange multipliers belong to some L p (Ω T ) for p > 1. Proof. Key are a priori estimates and regularity for y and ρ and Lagrange multiplier theorem. Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 10 / 17

18 Analytic results Theorem (Existence) For δ, ε > 0, there exists at least one minimum and the corresponding Lagrange multipliers belong to some L p (Ω T ) for p > 1. Proof. Key are a priori estimates and regularity for y and ρ and Lagrange multiplier theorem. Passing to the limit for ε, δ 0? Necessary condition for convergence of the whole system is δ ε. Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 10 / 17

19 Case ε δ: parasitic currents min δ ρ W (ρ) s.t. (NSE ε ). Ω T δ Ω T ρ(t = 0) ρ(t = 0.25) ρ(t = 0.5) Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 11 / 17

20 Case ε δ: parasitic currents min δ ρ W (ρ) s.t. (NSE ε ). Ω T δ Ω T ρ(t = 0) ρ(t = 0.25) ρ(t = 0.5) y(t = 0.05) y(t = 0.15) y(t = 0.35) Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 11 / 17

21 Case ε δ: massive diffusion min δ ρ W (ρ) s.t. (NSE ε ). Ω T δ Ω T ρ(t = 0) ρ(t = 0.5) moderate ε ρ(t = 0.5) big ε Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 12 / 17

22 Contents 1 Introduction and Motivation 2 Analysis 3 Numerical Analysis and Computations Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 13 / 17

23 Strategy Strategy for the discretization Fix δ, ε > 0. Use first discretize, then optimize ansatz with convergent and unconditionally stable scheme, cf. [Ba nas and Prohl, 2010]. Show existence of discrete optimum, derive discrete optimality conditions. Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 14 / 17

24 Strategy Strategy for the discretization Fix δ, ε > 0. Use first discretize, then optimize ansatz with convergent and unconditionally stable scheme, cf. [Ba nas and Prohl, 2010]. Show existence of discrete optimum, derive discrete optimality conditions. Strong coupling of primal and dual variables in the adjoint equation Need to bound strong norms of the primal variables Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 14 / 17

25 Strategy Strategy for the discretization Fix δ, ε > 0. Use first discretize, then optimize ansatz with convergent and unconditionally stable scheme, cf. [Ba nas and Prohl, 2010]. Show existence of discrete optimum, derive discrete optimality conditions. Strong coupling of primal and dual variables in the adjoint equation Need to bound strong norms of the primal variables, in particular show: sup [ Y(t) 2 + h R(t) 2] t [0,T ] T + h Y(t) 2 + d t Y(t) 2 + d t R(t) 2 dt C. 0 Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 14 / 17

26 Strategy Strategy for the discretization Fix δ, ε > 0. Use first discretize, then optimize ansatz with convergent and unconditionally stable scheme, cf. [Ba nas and Prohl, 2010]. Show existence of discrete optimum, derive discrete optimality conditions. Strong coupling of primal and dual variables in the adjoint equation Need to bound strong norms of the primal variables, in particular show: sup [ Y(t) 2 + h R(t) 2] t [0,T ] T + h Y(t) 2 + d t Y(t) 2 + d t R(t) 2 dt C. 0 Bounds for dual variables. Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 14 / 17

27 Main result Theorem (Convergence) There exist y, p, ρ; z, q, η; u : Ω T R (2), such that the solutions of the fully discrete optimality system converge to them in some norms (up to subsequences). The limit functions solve the original fully continuous optimality system. Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 15 / 17

28 Summary Done New geometric functional considered with PDE constraints: Evidence, existence and optimality conditions for δ, ε > 0. Rigorous converence analysis with unconditionally stable scheme for δ, ε > 0. Implementation for δ, ε > 0. Outlook What happens for ε, δ 0? Proofs? Interplay between δ, ε and numerical parameters (time step size k and grid size h)? Surface tension instead of geometric functional? Other models (sharp interface, thin film, etc.)? Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 16 / 17

29 Summary Done New geometric functional considered with PDE constraints: Evidence, existence and optimality conditions for δ, ε > 0. Rigorous converence analysis with unconditionally stable scheme for δ, ε > 0. Implementation for δ, ε > 0. Outlook What happens for ε, δ 0? Proofs? Interplay between δ, ε and numerical parameters (time step size k and grid size h)? Surface tension instead of geometric functional? Other models (sharp interface, thin film, etc.)? Thank you for your attention! Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 16 / 17

30 References I Ba nas, L. and Prohl, A. (2010). Convergent finite element discretization of the multi-fluid nonstationary incompressible magnetohydrodynamics equations. Math. Comp., 79(272): Braides, A. (1998). Approximation of free discontinuity problems. Number 1694 in Lecture notes in mathematics. Springer, Berlin. Gerbeau, J.-F., Le Bris, C., and Lelièvre, T. (2006). Mathematical methods for the magnetohydrodynamics of liquid metals. Numerical mathematics and scientific computation. Oxford University Press. Kunisch, K. and Lu, X. (2011). Optimal control for multi-phase fluid stokes problems. Nonlinear Anal., 74(2): Lions, P.-L. (1996). Mathematical topics in fluid mechanics, volume 1: Incompressible models of Oxford lecture series in mathematics and its applications. Clarendon Press. Markus Klein (U Tübingen) Control of Interface Evolution in Multi-Phase Flows Garching 17 / 17