Optimal control in fluid mechanics by finite elements with symmetric stabilization

Transcription

1 Computational Sciences Center Optimal control in fluid mechanics by finite elements with symmetric stabilization Malte Braack Mathematisches Seminar Christian-Albrechts-Universität zu Kiel VMS Worshop 2008 Saarbrücken, June, / 29

2 Menu 1 Motivation 2 Finite element discretization 3 Finite elements with symmetric stabilization 4 A convergence result 5 Examples of symmetric stabilization techniques 6 Numerical validation 2 / 29

3 Menu 1 Motivation 2 Finite element discretization 3 Finite elements with symmetric stabilization 4 A convergence result 5 Examples of symmetric stabilization techniques 6 Numerical validation 2 / 29

4 1. Motivation Optimization problem Build Karush-Kuhn-Tucker (KKT) system Solve KKT system 3 / 29

5 Two possibilities for optimization with PDE Optimization problem with PDE Build KKT system Discretize PDE Discretize KKT system Build discretize KKT system Solve discrete KKT system 4 / 29

6 Two possibilities for optimization with PDE Optimization problem with PDE Build KKT system Discretize PDE Discretize KKT system Build discretize KKT system Solve discrete KKT system Optimize-discretize Discretize-optimize 4 / 29

7 Model problem: Linearized Navier-Stokes with control q µ v + (β )v + σv + p + Bq = f in Ω, div v = 0 in Ω, v = 0 on Ω, Objective functional: J(u, q) := 1 2 Cu Cû 2 + α 2 q 2 min! Cû = v observationes 5 / 29

8 Linear flow problem: Au + Bq = f state variable u = (v, p), and control q Optimal control problem: { } arg min J(u, q) : Au + Bq = f for control q Q. 6 / 29

9 Augmented Lagrangian L(u, q, z) := J(u, q) + z, Au + Bq f Unrestricted minimization problem min u,q,z L(u, q, z) Necessary conditions for saddle point of L d q L(u, q, z) = 0 d q J(u, q) + B z = 0 d u L(u, q, z) = 0 d u J(u, q) + A z = 0 d z L(u, q, λ) = 0 Au + Bq = f 7 / 29

10 Continuous Karush-Kuhn-Tucker (KKT) system αi 0 B 0 C A B A 0 q u = z 0 Cû f 8 / 29

11 What is an appropriate discretization of... Primal equation µ v + (β )v + σv + p + Bq = f in Ω div v = 0 in Ω v = 0 on Ω Adjoint equation µ z v (β )z v + σz v z p = v v in Ω div z v = 0 in Ω z v = 0 on Ω 9 / 29

12 2. Finite element discretization Bilinear form for u = (v, p) X := [H0 1(Ω)]d L 2 0 (Ω) a(u, ϕ) := (div v, ξ) + (σv, φ) + (β v, φ) + (µ v, φ) (p, div φ) Influence of the control by b : Q X R for q Q L 2 (Ω). Variational formulation: u X : a(u, ϕ) + b(q, ϕ) = (f, ϕ) ϕ X Galerkin formulation: u h X h : a(u h, ϕ) + b(q h, ϕ) = (f, ϕ) ϕ X h 10 / 29

13 SUPG+PSPG, Grad-div stabilization for Oseen Inf-sup condition not fulfilled for equal-order elements Dominant convective terms s h (u h )(ϕ) = {ρ mom [δ T (β )φ + α T ξ] + (div v) γ T (div φ)} dx T T h T (Hughes, Johnson, Lube, Tobiska, Glowinski, Le Tallec,..) 11 / 29

14 SUPG+PSPG, Grad-div stabilization for Oseen Inf-sup condition not fulfilled for equal-order elements Dominant convective terms s h (u h )(ϕ) = {ρ mom [δ T (β )φ + α T ξ] + (div v) γ T (div φ)} dx T T h T (Hughes, Johnson, Lube, Tobiska, Glowinski, Le Tallec,..) Discretized primal problem: (A h + S u h )u h + (B h + S q h )q h = f h Forget for a while the parameter dependence: Sh u, S q h Otherwise: Sh u, S q h, f h may depend on u h. are linear. 11 / 29

15 Adjoint equation: µ z v (β )z v + σz v z p = v v in Ω div z v = 0 in Ω z v = 0 on Ω is also of Ossen type and need to be stabilized. Discretized adjoint problem: (A h + S z h )z h + Cu h = Cû For residual based stabilization: S z h depend on the full adjoint residual. 12 / 29

16 Discrete KKT system (optimize-discretize): αi 0 B h 0 C h A h + S z h B h + S q h A h + S u h 0 q h u h = z h 0 Cû f h The other way round (discretize-optimize): cf. Collis & Heinkenschloss [2002] Build KKT system of discretized PDE: (A h + S u h )u + (B h + S q h )q h = f h αi 0 B h + (S q h ) 0 C h A h + (S u h ) B h + S q h A h + S u h 0 q h u h = z h 0 Cû f h In general: S q h 0 and S z h (S u h ). 13 / 29

17 Streamline diffusion & pressure stabilized Petrov Galerkin (S u h ) S z h K + K {( v h v h + σz v + (β )z v µ z v, δ p ξ) K } {(σφ + (β )φ µ φ, δ p z p ) K } ( v h v h z p, δ v (β )φ) + ( ξ, δ v (β )z v ) Numerical tests by Collis & Heinkenschloss [2002]: D-O has better convergence properties than O-D for SUPG; large differences in z h between D-O and O-D. 14 / 29

18 From Abraham, Behr, Heinkenschloss (2004): GLS comparison of do and od with different settings of stabilization constants: diag: h K :=max. element lenght adv: h K := i (β K )φ i K / β K, (Tezduyar, Park (1986)) 15 / 29

19 3. Finite elements with symmetric stabilization Consider linear stabilization: a(u h, ϕ) + b(q h, ϕ) + s h (u h, ϕ) = (f, ϕ) ϕ X h First requirement Symmetry: (P1) s h (u, ϕ) = s h (ϕ, u) u, ϕ X Lemma: For linear and symmetric stabilization (P1), discretization and optimization commutes. 16 / 29

20 We will show an a priori estimate in a (semi) norm: Second requirement Coercivity: h : X R + 0 (P2) u h 2 h a h (u h, u h ) + s h (u h, u h ) u h X h This is the case e.g. for if s h (u, u) 0. u h := (a h (u, u) + s h (u, u)) 1/2 17 / 29

21 Third requirement: u h h stronger than L 2 -norm of velocities: (P3) v u h u = (v, p) X For example: u 2 h = σ v 2 + µ v 2 + s h (u, u) 18 / 29

22 Fourth requirement: a priori estimate for fixed control. For u [H r+1 (Ω)] d+1 and finite elements of order r: (P4) u(q) u h (q) h h s u r+1 u(q), u h (q) = solutions of continuous and discrete problems for given control q Q. convergence order s r + 1 (optimal s = r + 1/2) Lemma: If (P4) holds for the primal problem, then it holds for the adjoint problems with given velocity field w in the rhs: z(w) z h (w) h h s z r+1 if z [H r+1 (Ω)] d+1 19 / 29

23 4. A convergence result Theorem Under the following conditions: (P1), (P2), (P3), (P4) approximation property of the discrete control space: q i h q h s q r+1 regularity of the solutions: u, z [H r+1 (Ω)] d+1, q H r+1 (Ω) it holds the convergence result: q q h h s ( u r+1 + z r+1 + q r+1 ) 20 / 29

24 Principle of proof: Since the reduced functional j h (q) := J(u h (q), q) is at most quadratic: α i h q q }{{ h 2 j } h (q h)(δq h ) = j h (q h + δq h )(δq }{{} h ) j h (q h)(δq h ) }{{} =:δq h =i h q =0=j (q)(δq h ) Expressing j and j h and continuity of b(, ) gives (ẑ h := z h (u h (i h q))): α i h q q h 2 b(i h q q h, ẑ v h zv ) + (α( i h q q), δq h ) c ẑ v h zv i h q q h + α i h q q i h q q h ẑh v zv zh v (u h(i h q)) zh v }{{ (u(q)) + zh v } (u(q)) zv (u(q)) }{{} stab. disc. adjoint & primal pb. prev. Lemma 21 / 29

25 Theorem Under the same conditions as the previous theorem with s = r : u u h 2 h h r+ 1 2 ( u r+1 + z r+1 + q r+1 ) Proof. u u h h u(q) u h (q) }{{ h + u } h (q) u h (q h ) h h r+ 1 2 u r+1 due to (P4) Coercivity (P2) for w h := u h (q) u h (q h ): w h 2 h a(w h, w h ) + s h (w h, w h ) = (B(q q h ), wh v ) Cauchy-Schwarz, (P3) and continuity of B: w h h B(q q h ) q q h 22 / 29

26 5. Examples of symmetric stabilization techniques Edge oriented stabilization (EOS) [Burman, Hansbo] Jumps across edges: [u(x)] := u(x) K u(x) K. K K Stabilization terms: s es h (u, ϕ) := ses,p h (p, ξ) + s es,v h (v, φ) s es,p h (p, ξ) := K T h s es,v h (v, φ) := K T h K K Fulfill (P1), (P2), (P3) and (P4). α K [ p] [ ξ] ds {δ K [n v] [n φ] + γk [div v] [div φ] } ds 23 / 29

27 5. Examples of symmetric stabilization techniques Edge oriented stabilization (EOS) [Burman, Hansbo] Jumps across edges: [u(x)] := u(x) K u(x) K. K K Stabilization terms: s es h (u, ϕ) := ses,p h (p, ξ) + s es,v h (v, φ) s es,p h (p, ξ) := K T h s es,v h (v, φ) := K T h K K α K [ p] [ ξ] ds {δ K [n v] [n φ] + γk [div v] [div φ] } ds Fulfill (P1), (P2), (P3) and (P4). Hence: optimal order of convergence. 23 / 29

28 Local projection stabilization (LPS) [Becker, Br., Burman, Tobiska, Matthies, Lube, Rapin] Step 1 - Definition of fluctuation operator: D r 1 2h = discontinuous, patchwise polynomial order r 1. T h T 2h Patchwise L 2 -projection Fluctuation operator π h : L 2 (Ω) D r 1 2h κ h = i π h Example r = 1: Patch-wise projection on constants: κ h p K = p 1 p dx, K T 2h K K 24 / 29

29 Step 2 - Definition of stabilization terms Pressure stabilization (Br. & Becker 00) ( ) S h (u, ϕ) = κ h ( p), ακ h ( ξ) stabilization of convective terms by the full gradient ( )... + κ h ( v), δκ h ( φ) or streamline derivatives + stabilization of divergence-free condition ( ) ( )... + κ h ((β )v), δκ((β )φ) + κ h (div v), γκ((div φ) But: nonlinear for Navier-Stokes. Fulfill (P1), (P2), (P3) and (P4). 25 / 29

30 Step 2 - Definition of stabilization terms Pressure stabilization (Br. & Becker 00) ( ) S h (u, ϕ) = κ h ( p), ακ h ( ξ) stabilization of convective terms by the full gradient ( )... + κ h ( v), δκ h ( φ) or streamline derivatives + stabilization of divergence-free condition ( ) ( )... + κ h ((β )v), δκ((β )φ) + κ h (div v), γκ((div φ) But: nonlinear for Navier-Stokes. Fulfill (P1), (P2), (P3) and (P4). Hence: optimal order of convergence. 25 / 29

31 6. Numerical validation Navier-Stokes: µ v + (v )v + p + Bq = f in Ω, div v = 0 in Ω, v = v 0 on Ω, Discretized with local projection stabilization. DFG benchmark: (uncontroled solution at Re = 100) 26 / 29

32 Objective functional: *x*(x-2.05) -0.3*(x-4.1)*(x-2.05) J(v, q) := 1 2 v v 2 min! v(x, y) =double-poiseulle flow (parabolic) Initial sol. optimiz. sol. 27 / 29

33 Comparison of convergence: 1 LPS GLS 0.1 L2 v-v_h e e+06 #nodes LPS = local projection stabilization (symmetric) GLS = PSPG / SUPG optimize-discretize 28 / 29

34 Comparison of convergence: 1 LPS GLS 0.1 L2 v-v_h e e+06 #nodes LPS = local projection stabilization (symmetric) GLS = PSPG / SUPG optimize-discretize Further optimization results with LPS: Becker, Meidner, Vexler 28 / 29

35 Summary Type of discretization is important for flow control. 29 / 29

36 Summary Type of discretization is important for flow control. Finite element schemes may provide consistent KKT systems when symmetric stabilization is used. 29 / 29

37 Summary Type of discretization is important for flow control. Finite element schemes may provide consistent KKT systems when symmetric stabilization is used. Convergence proof for Oseen with general symmetric stabilization (LPS,EOS,...) 29 / 29

38 Summary Type of discretization is important for flow control. Finite element schemes may provide consistent KKT systems when symmetric stabilization is used. Convergence proof for Oseen with general symmetric stabilization (LPS,EOS,...) First numerical test problem indicate the benefit of symmetric stabilization. 29 / 29

39 Summary Type of discretization is important for flow control. Finite element schemes may provide consistent KKT systems when symmetric stabilization is used. Convergence proof for Oseen with general symmetric stabilization (LPS,EOS,...) First numerical test problem indicate the benefit of symmetric stabilization. Thanks a lot! 29 / 29