Flow control in the presence of shocks Enrique Zuazua BCAM - Basque Center for Applied Mathematics & Ikerbasque Bilbao, Basque Country, Spain zuazua@bcamath.org http://www.bcamath.org/zuazua/ PDE’s, Dispersion, Scattering theory and Control theory, Monastir, Tunisia, June 2013 Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 1 / 59
Outline 1 Introduction: Motivation and examples 2 Optimal shape design in aeronautics 3 Shocks: Some remedies 4 Related topics Several space dimensions (R. Lecaros) Large time asymptotics (L. Ignat & A. Pozo) Steady state models (M. Ersoy, E. feireisl & E. Z.) Viscous models Flux identification 5 Perspectives Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 2 / 59
Intro Outline 1 Introduction: Motivation and examples 2 Optimal shape design in aeronautics 3 Shocks: Some remedies 4 Related topics Several space dimensions (R. Lecaros) Large time asymptotics (L. Ignat & A. Pozo) Steady state models (M. Ersoy, E. feireisl & E. Z.) Viscous models Flux identification 5 Perspectives Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 3 / 59
Intro Motivation Control problems for PDE are interesting for at least two reasons: They emerge in most real applications. PDE as the models of Continuum and Quantum Mechanics. Furthermore, in real world, there is something to be optimized, controlled, optimally shaped, etc. Answering to these control problems often requires a deep understanding of the underlying dynamics and a better master of the standard PDE models. Surprisingly enough, this has led to an important ensemble of new tools and results and some fascinating problems are still widely open. Furthermore, these kind of techniques are of application in some other fields, such as inverse problems theory and parameter identification issues. Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 4 / 59
Intro The same can be said about the numerical approximation aspects of these problems. Classical numerical analysis techniques do not suffice. Furthermore, from a control theoretical viewpoint this raises interesting issues about passing to the limit from finite to infinite space dimensions. Discrete versus continuous approaches.... In this talk we discuss the impact of shock discontinuities in solutions when addressing control problems for some toy models in Fluid Mechanics. Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 5 / 59
Intro Some relevant applications Noise reduction in cavities and vehicles. “Acoustic noise reduction” versus “active versus passive controllers”. Laser control in quantum mechanical and molecular systems. Seismic waves, earthquakes. Flexible structures. Environment: The Thames Barrier. Optimal shape design in aeronautics. Human cardiovascular system. Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 6 / 59
Intro Introduction A complete analysis of these issues involves, at least: Partial Differential Equations: Models describing motion in the various fields of Mechanics: Elasticity, Fluids,... Numerical Analysis: Allowing to discretize these models so that solutions may be approximated algorithmically. Optimal Design: Design of shapes to enhance the desired properties (bridges, dams, airplanes,..) Control: Automatic and active control of processes to guarantee their best possible behavior and dynamics. Parameter identification, inverse problems. Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 7 / 59
Intro Computer simulation → far beyond the fields in which its use is justified (consistency + stability ≡ convergence). The risk: To end up getting numerical data whose validity.... Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 8 / 59
Intro Is this difficulty solvable in practice? Solvable for problems with well known data by means of post-processing. Much harder for inverse, design and control problems,,,, In those cases the obtained final numerical results and simulations may simply mean nothing. Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 9 / 59
Shape design in aeronautics Outline 1 Introduction: Motivation and examples 2 Optimal shape design in aeronautics 3 Shocks: Some remedies 4 Related topics Several space dimensions (R. Lecaros) Large time asymptotics (L. Ignat & A. Pozo) Steady state models (M. Ersoy, E. feireisl & E. Z.) Viscous models Flux identification 5 Perspectives Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 10 / 59
Shape design in aeronautics Shape design in aeronautics Optimal shape design in aeronautics. Two aspects: Shocks. Oscillations. Optimal shape ∼ Active control. The shape of the cavity or airfoil controls the surrounding flow of air. Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 11 / 59
Shape design in aeronautics Optimal shape design in aeronautics Aeronautics: to simulate and optimize complex processes is indispensable. Long tradition: J. L. Lions, A. Jameson,... However, this needs an immense computational effort. For practical optimization problems, in which at least 100 design variables are to be considered, current methodological approaches applied in industry will need more than a year to obtain an optimized aircraft. Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 12 / 59
Shape design in aeronautics Francisco Palacios, Stanford University Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 13 / 59
Shape design in aeronautics Mathematical problem formulation Minimize J (Ω ∗ ) = min J (Ω) Ω ∈ C ad C ad = class of admissible domains. J = cost functional (drag reduction, lift maximization, exploitation cost, overall cost over the life cycle of the aircraft, benefit maximization, etc). J depends on Ω through u (Ω), solution of the PDE (elasticity, Fluid Mechanics,...). The domains under consideration are often complex. Geometric and parametrization issues play a key role. Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 14 / 59
Shape design in aeronautics The dependence of the functional on the domain, through the solution of the PDE is complex as well. J it is far from being a nice convex function. Analytical difficulties: Lack of good existence, uniqueness, and continuous dependence theory for the PDE. Lack of convexity of the functional. Lack of compactness within the class of relevant domains... Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 15 / 59
Shape design in aeronautics In practice Descent algorithm (gradient based method) on a discrete version of the problem: The domains Ω have been discretized (finite element mesh) The PDE has been replaced by a numerical scheme, The functional J has been replaced by a discrete version. Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 16 / 59
Shape design in aeronautics Note however that computing gradients, in practice, may be very hard. Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 17 / 59
Shape design in aeronautics We end up with a discrete optimization problem of huge dimensions. Divergence of numerical iterative algorithms may be hard to detect. [Boundary control of vibrations, E. Z. SIAM Review, 2005] Can we guarantee this kind of pathologies do not arise in realistic problems of optimal shape design in aeronautics? Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 18 / 59
Shape design in aeronautics Two approaches: Discrete: Discretization + gradient Advantages: Discrete clouds of values. No shocks. Automatic differentiation, ... Drawbacks: ”Invisible” geometry. Scheme dependent. Continuous: Continuous gradient + discretization. Advantages: Simpler computations. Solver independent. Shock detection. Drawbacks: Yields approximate gradients. Subtle if shocks. Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 19 / 59
Shocks: A remedy Outline 1 Introduction: Motivation and examples 2 Optimal shape design in aeronautics 3 Shocks: Some remedies 4 Related topics Several space dimensions (R. Lecaros) Large time asymptotics (L. Ignat & A. Pozo) Steady state models (M. Ersoy, E. feireisl & E. Z.) Viscous models Flux identification 5 Perspectives Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 20 / 59
Shocks: A remedy The relevant models in aeronautics (Fluid Mechanics): Navier-Stokes equations; Euler equations; Turbulent models: Reynolds-Averaged Navier-Stokes (RANS), Spalart-Allmaras Turbulence Model, k − ε model; .... Burgers equation (as a 1 − d theoretical laboratory). Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 21 / 59
Shocks: A remedy Solutions may develop shocks or quasi-shock configurations. For shock solutions, classical calculus fails; For quasi-shock solutions the sensitivity is so large that classical sensitivity clalculus is meaningless. Enrique Zuazua (BCAM) Flow control in the presence of shocks Monastir, June 2013 22 / 59
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