×

Optimal control of flow with discontinuities. (English) Zbl 1061.76076

Summary: Optimal control of the 1-D Riemann problem of Euler equations is studied, with the initial values for pressure and density as control parameters. The least-squares type cost functional employs either distributed observations in time or observations calculated at the end of the assimilation window. Existence of solutions for the optimal control problem is proven. Smooth and nonsmooth optimization methods employ the numerical gradient (respectively, a subgradient) of the cost functional, obtained from the adjoint of the discrete forward model. The numerical flow obtained with the optimal initial conditions obtained from the nonsmooth minimization matches very well with the observations. The algorithm for smooth minimization converges for the shorter time horizon but fails to perform satisfactorily for the longer time horizon, except when the observations corresponding to shocks are detected and removed.

MSC:

76N25 Flow control and optimization for compressible fluids and gas dynamics
49J20 Existence theories for optimal control problems involving partial differential equations
65K10 Numerical optimization and variational techniques
76M30 Variational methods applied to problems in fluid mechanics

Software:

NDA; PNEW; L-BFGS; CLAWPACK
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Addington, G. A.; Hyatt, J. M., Control-surface deflection effects on aerodynamic response nonlinearities, AIAA Paper, 4107 (2000)
[2] Anderson, W. K.; Venkatakrishnan, A., Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation, Comput. Fluids, 28, 443 (1999) · Zbl 0968.76074
[3] J.D. Anderson, Fundamentals of aerodynamics, McGraw-Hill Series in Aeronautical and Aerospace Engineering, McGraw-Hill, 2001, 912 pp; J.D. Anderson, Fundamentals of aerodynamics, McGraw-Hill Series in Aeronautical and Aerospace Engineering, McGraw-Hill, 2001, 912 pp
[4] E. Arian, M.D. Salas, Admitting the inadmissible: adjoint formulation for incomplete cost functionals in aerodynamic optimization, ICASE Report, vols. 97-69, 1997; E. Arian, M.D. Salas, Admitting the inadmissible: adjoint formulation for incomplete cost functionals in aerodynamic optimization, ICASE Report, vols. 97-69, 1997
[5] Bardos, C.; Pironneau, O., A formalism for the differentiation of conservation laws, C. R. Acad. Sci. Paris, 335, 10, 839 (2000) · Zbl 1020.35052
[6] Bardos, C.; Pironneau, O., Derivatives and control in the presence of shocks, CFD J., 12, 1 (2003)
[7] Barton, P. I.; Banga, J. R.; Galan, S., Optimization of hybrid discrete continuous dynamic systems, Comput. Chem. Engrg., 24, 2171 (2000)
[8] Bein, T.; Hanselka, H.; Breitbach, E., An adaptive spoiler to control the transonic shock, Smart Mater. Struct., 9, 2, 141 (2000)
[9] Birkmeyer, J.; Rosemann, H.; Stanewsky, E., Shock control on a swept wing, Aerosp. Sci. Technol., 4, 3, 147 (2000) · Zbl 0952.76536
[10] Birta, L. G.; Oren, T. I., A robust procedure for discontinuity handling in continuous system simulation, T. Soc. Comput. Simul., 2, 3, 189 (1985)
[11] Bonnans, J. F.; Gilbert, J.; Lemarechal, C.; Sagastizabal, C., Optimisation Numerique: Aspects Theoriques et Pratiques (1997), Springer: Springer Paris · Zbl 0952.65044
[12] A. Bressan, Hyperbolic systems of conservation laws in one space dimension, in: Proceedings International Congress of Mathematicians, Beijing, 2002; A. Bressan, Hyperbolic systems of conservation laws in one space dimension, in: Proceedings International Congress of Mathematicians, Beijing, 2002 · Zbl 1032.35129
[13] Bressan, A., Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem (2000), Oxford University Press: Oxford University Press Oxford, (264 p.) · Zbl 0997.35002
[14] Bressan, A.; Crasta, G.; Piccoli, B., Well-posedness of the Cauchy problem for \(n\)×\(n\) systems of conservation laws, Memoirs A.M.S., 694 (2000) · Zbl 0958.35001
[15] Cacuci, D. G., Global optimization and sensitivity analysis, Nucl. Sci. Engrg., 104, 78 (1990)
[16] Cliff, E. M.; Heinkenschloss, M.; Shenoy, A. R., Adjoint-based methods in aerodynamic design optimization, (Computational Methods for Optimal Design and Control (1996), Birkhauser: Birkhauser Basel), 91 · Zbl 0988.76083
[17] E.M. Cliff, M. Heinkenschloss, A.R. Shenoy, On the optimality system for a 1-D Euler flow problem, AIAA Paper 96-3993 (1996); E.M. Cliff, M. Heinkenschloss, A.R. Shenoy, On the optimality system for a 1-D Euler flow problem, AIAA Paper 96-3993 (1996)
[18] Cliff, E. M.; Heinkenschloss, M.; Shenoy, A. R., An optimal control problem for flows with discontinuities, J. Optim. Theory Appl., 94, 2, 273 (1997) · Zbl 0891.49018
[19] T.J. Cowan, private communication, 2001; T.J. Cowan, private communication, 2001
[20] Dadone, A.; Valorani, M.; Grossman, B., Optimization of 2D fluid design problems with nonsmooth or noisy objective function, (Computational Fluid Dynamics ’96 (1996), Wiley: Wiley New York), 425
[21] Dadone, A.; Grossman, B., Progressive optimization of inverse fluid dynamic design problems, Comput. Fluids, 29, 1 (2000) · Zbl 0955.76081
[22] A. Dadone, B. Grossman, Fast convergence of inviscid fluid dynamic design problems, in: Proceedings of ECCOMAS 2000, Barcelona, September 11-14, 2000, Available from www.imamod.ru/jour/conf/ECCOMAS2000/pdf/148.pdf; A. Dadone, B. Grossman, Fast convergence of inviscid fluid dynamic design problems, in: Proceedings of ECCOMAS 2000, Barcelona, September 11-14, 2000, Available from www.imamod.ru/jour/conf/ECCOMAS2000/pdf/148.pdf · Zbl 0955.76081
[23] Fletcher, R., Practical Methods of Optimization (1987), Wiley: Wiley New York, 450 pp · Zbl 0905.65002
[24] Frank, P. D.; Shubin, G. Y., A comparison of optimization-based approaches for a model computational aerodynamics design problem, J. Comput. Phys., 98, 74 (1992) · Zbl 0741.76067
[25] M.B. Giles, N.A. Pierce, Adjoint equations in CFD: duality, boundary conditions and solution behaviour, AIAA Paper 97-1850 (1997); M.B. Giles, N.A. Pierce, Adjoint equations in CFD: duality, boundary conditions and solution behaviour, AIAA Paper 97-1850 (1997)
[26] Giles, M. B.; Pierce, N. A., On the properties of solutions of the adjoint Euler equations, (Numerical Methods for Fluid Dynamics VI (1998), ICFD: ICFD Oxford)
[27] Giles, M. B.; Pierce, N. A., Analytic adjoint solutions for the quasi-1D Euler equations, J. Fluid Mech., 426, 327 (2001) · Zbl 0967.76079
[28] Giles, M. B.; Pierce, N. A., An introduction to the adjoint approach to design, Flow Turbul. Combust., 65, 3-4, 393 (2000) · Zbl 0996.76023
[29] M.B. Giles, Discrete adjoint approximations with shocks, in: Proceedings of HYP2002 Conference, Caltech, March 2002; M.B. Giles, Discrete adjoint approximations with shocks, in: Proceedings of HYP2002 Conference, Caltech, March 2002 · Zbl 1134.76396
[30] Godlewski, E.; Raviart, P. A., Numerical Approximation of Hyperbolic Systems of Conservation Laws (1996), Springer: Springer New York, 508 pp · Zbl 0860.65075
[31] Gunzburger, M., Sensitivities, adjoints and flow optimization, Int. J. Numer. Methods Fluid, 31, 53 (1999) · Zbl 0962.76030
[32] Gunzburger, M., Sensitivities in computational methods for optimal flow control, (Computational Methods for Optimal Design and Control (1996), Birkhauser: Birkhauser Basel), 197 · Zbl 0988.76084
[33] Habbal, A., Direct approach to the minimization of the maximal stress over an arch structure, J. Optim. Theory Appl., 97, 3, 551 (1998) · Zbl 0914.73038
[34] Habbal, A., Nonsmooth shape optimization applied to linear acoustics, SIAM J. Optim., 8, 4, 989 (1998) · Zbl 0922.65050
[35] Hassold, E., Automatic differentiation applied to a nonsmooth optimization problem, (Numerical Methods in Engineering ’96 (1996), Wiley: Wiley New York), 835
[36] Homescu, C.; Navon, I. M.; Li, Z., Suppression of vortex shedding for flow around a circular cylinder using optimal control, Int. J. Numer. Methods Fluids, 38, 43 (2002) · Zbl 1007.76019
[37] Homescu, C.; Navon, I. M., Numerical and theoretical considerations for sensitivity calculation of discontinuous flow, Syst. Control Lett., 48, 253 (2003) · Zbl 1157.93361
[38] L. Huyse, M.R. Lewis, Aerodynamic shape optimization of two-dimensional airfoils under uncertain conditions, ICASE Report, vol. 2001-1, 2001; L. Huyse, M.R. Lewis, Aerodynamic shape optimization of two-dimensional airfoils under uncertain conditions, ICASE Report, vol. 2001-1, 2001
[39] Iollo, A.; Salas, M. D., Optimum transonic airfoils based on Euler equations, Comput. Fluids, 28, 653 (1999) · Zbl 0956.76038
[40] A. Iollo, M.D. Salas, S. Ta’asan, Shape optimization governed by the Euler equations using an adjoint method, ICASE Report, vols. 93-78, 1993; A. Iollo, M.D. Salas, S. Ta’asan, Shape optimization governed by the Euler equations using an adjoint method, ICASE Report, vols. 93-78, 1993
[41] Iollo, A.; Salas, M. D., Contribution to the optimal shape design of two-dimensional internal flows with embedded shocks, J. Comput. Phys., 125, 1, 124 (1996) · Zbl 0848.76072
[42] James, F.; Sepulveda, M., Convergence results for the flux identification in a scalar conservation law, SIAM J. Control Optim., 37, 3, 869 (1999) · Zbl 0970.35161
[43] Jameson, A., A perspective on computational algorithms for aerodynamic analysis and design, Prog. Aerosp. Sci., 37, 197 (2001)
[44] Lee, D.; Pavlidis, T., One-dimensional regularization with discontinuities, IEEE Trans. Pattern Anal. Machine Intell., 10, 6, 822 (1988) · Zbl 0657.65020
[45] Lemarechal, C., Nondifferentiable optimization, subgradient and \(ϵ\) subgradient methods, (Optimization and Operations Research. Optimization and Operations Research, Lecture Notes in Economics and Mathematical Systems, 117 (1976), Springer: Springer Berlin), 191 · Zbl 0358.90066
[46] Lemarechal, C., Nondifferentiable optimization, (Handbooks in Operations Research and Management Science, vol. 1. Optimization (1989), North-Holland: North-Holland Amsterdam) · Zbl 0591.90080
[47] Leonard, D.; Long, N. V., Optimal Control Theory and Static Optimization in Economics (1992), Cambridge University Press: Cambridge University Press Cambridge, 368 pp
[48] Leveque, R., Numerical Methods for Conservation Laws (1992), Birkhauser: Birkhauser Basel · Zbl 0847.65053
[49] CLAWPACK: A Software Package for Conservation Laws and Hyperbolic Systems, Available from http://www.amath.washington.edu/ rjl/clawpack; CLAWPACK: A Software Package for Conservation Laws and Hyperbolic Systems, Available from http://www.amath.washington.edu/ rjl/clawpack
[50] Liepmann, H. W.; Roshko, A., Elements of Gas Dynamics (1957), Wiley: Wiley New York · Zbl 0078.39901
[51] Liu, D.; Nocedal, J., On the limited memory BFGS method for large scale optimization, Math. Program. B, 45, 503 (1989) · Zbl 0696.90048
[52] Lohner, R.; Morgan, K.; Peraire, J.; Vahdati, M., Finite element flux-corrected transport (FEM-FCT) for the Euler and Navier-Stokes equations, Int. J. Numer. Methods Fluids, 7, 1093 (1987) · Zbl 0633.76070
[53] L. Luksan, J. Vlcek, NDA: algorithms for nondifferentiable optimization, ICS Report, vol. 797, Institute of Computer Science, Academy of Sciences of the Czech Republic, 2000; L. Luksan, J. Vlcek, NDA: algorithms for nondifferentiable optimization, ICS Report, vol. 797, Institute of Computer Science, Academy of Sciences of the Czech Republic, 2000 · Zbl 1070.65552
[54] Luksan, L.; Vlcek, J., Algorithm 811: NDA: algorithms for nondifferentiable optimization, ACM T. Math. Software, 27, 2, 193 (2001) · Zbl 1070.65552
[55] Makela, M. M.; Neittaanmaki, P., Nonsmooth Optimization (1992), World Scientific: World Scientific Singapore · Zbl 0748.90065
[56] Mao, G.; Petzold, L. R., Efficient integration over discontinuities for differential-algebraic systems, Comput. Math. Appl., 43, 65 (2002) · Zbl 0999.65083
[57] Matsuzawa, T.; Hafez, M., Optimum shape design using adjoint equations for compressible flows with shock waves, Int. J. Comput. Fluid D, 7, 3, 343 (1998)
[58] Matsuzawa, T.; Hafez, M., Treatment of shock waves in design optimization via adjoint equation approach, Int. J. Comput. Fluid D, 7, 4, 405 (1998)
[59] Mohammadi, B.; Pironneau, O., Applied Shape Optimization for Fluids (2001), Oxford Science Publications: Oxford Science Publications Oxford, 368 pp · Zbl 0970.76003
[60] Narducci, R. P.; Grossman, B.; Haftka, R. T., Sensitivity algorithms for an inverse design problem involving a shock wave, Inverse Probl. Engrg., 2, 49 (1995)
[61] Navon, I. M.; Zou, X.; Derber, J.; Sela, J., Variational data assimilation with an adiabatic version of the NMC spectral model, Mon. Weather Rev., 120, 1433 (1992)
[62] Nocedal, J., Updating quasi-Newton matrices with limited storage, Math. Comput., 151, 35, 773 (1980) · Zbl 0464.65037
[63] Nocedal, J.; Wright, S. J., Numerical Optimization (1999), Springer: Springer Berlin, 600 pp · Zbl 0930.65067
[64] A. Oyama, S. Obayashi, K. Nakahashi, Transonic wing optimization using genetic algorithm, AIAA Paper 97-1854 (1997); A. Oyama, S. Obayashi, K. Nakahashi, Transonic wing optimization using genetic algorithm, AIAA Paper 97-1854 (1997)
[65] Park, T.; Barton, P. I., State event location in differential-algebraic models, ACM Trans. Model. Comput. Simul., 6, 2, 137 (1996) · Zbl 0887.65075
[66] Schramm, H.; Zowe, J., A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results, SIAM J. Optim., 2, 1, 121 (1992) · Zbl 0761.90090
[67] Shakib, F.; Hughes, T. J.R.; Johan, Z., A new finite formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equation, Comput. Methods Appl. Mech. Engrg., 89, 141 (1991) · Zbl 0838.76040
[68] Sod, G. A., A survey of finite-difference methods for systems of nonlinear conservation laws, J. Comput. Phys., 27, 1, 1 (1978) · Zbl 0387.76063
[69] Stanewsky, E., Adaptive wing and flow control technology, Prog. Aerosp. Sci., 37, 583 (2001)
[70] Tolsma, J. E.; Barton, P. L., Hidden discontinuities and parametric sensitivity calculations, SIAM J. Sci. Comput., 23, 6, 1862 (2002)
[71] Valorani, M.; Dadone, A., Sensitivity derivatives for non-smooth or noisy objective functions in fluid design problems, (Numerical Methods for Fluid Dynamics, vol. 5 (1995), Oxford Clarendon Press), 605 · Zbl 0871.76068
[72] Ulbrich, S., On the existence and approximation of solutions for the optimal control of nonlinear hyperbolic conservation laws, (Optimal Control of Partial Differential Equations (Chemnitz 1998). Optimal Control of Partial Differential Equations (Chemnitz 1998), International Series of Numerical Mathematics, 133 (1999), Birkhauser: Birkhauser Basel), 287 · Zbl 0933.49002
[73] Ulbrich, S., A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms, SIAM J. Control Optim., 41, 3, 740 (2002) · Zbl 1019.49026
[74] S. Ulbrich, Optimal control of nonlinear hyperbolic conservation laws with source terms, Habilitation Thesis, Fakultat fur Mathematik, Technische Universitat Munchen, January 2002; S. Ulbrich, Optimal control of nonlinear hyperbolic conservation laws with source terms, Habilitation Thesis, Fakultat fur Mathematik, Technische Universitat Munchen, January 2002
[75] Ulbrich, S., Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws, Syst. Control Lett., 48, 3-4, 309 (2002)
[76] Vlcek, J.; Luksan, L., Globally convergent variable metric method for nonconvex nondifferentiable unconstrained minimization, J. Optim. Theory Appl., 111, 2, 407 (2001) · Zbl 1029.90060
[77] Wang, Y. J.; Matsuhisa, H.; Honda, Y., Loads on lower limb joints and optimal action of muscles for shock reduction, JSME Int. J. C-Mech. SY., 42, 3, 574 (1999)
[78] Xu, Q., Generalized adjoint for physical processes with parameterized discontinuities. Part I: Basic issues and heuristic examples Part II: Vector formulations and matching conditions, J. Atmos. Sci., 53, 8, 1123 (1996)
[79] Zhang, S.; Zou, X.; Ahlquist, J. E.; Navon, I. M.; Sela, J., Use of differentiable and nondifferentiable optimization algorithms for variational data assimilation with discontinuous cost functions, Mon. Weather Rev., 128, 12, 4031 (2000)
[80] Zhang, S.; Zou, X.; Ahlquist, J. E., Examination of numerical results from tangent linear and adjoint of discontinuous nonlinear models, Mon. Weather Rev., 129, 2791 (2001)
[81] Zou, X.; Navon, I. M.; Berger, M.; Phua, P. K.; Schlick, T.; LeDimet, F. X., Numerical experience with limited-memory quasi-Newton methods for large-scale unconstrained nonlinear minimization, SIAM J. Optim., 3, 3, 582 (1993) · Zbl 0784.90086
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.