Numerical solution of a flow-control problem: Vorticity reduction by dynamic boundary action

SIAM Journal on Scientific Computing

Volumn 19, Issue 3, 1998, Pages 829-860

  Berggren, Martin ^a,b  

^a RICE UNIVERSITY   (United States)

^b Aeronautical Research Institute of Sweden   (Sweden)

Author keywords

Adjoint equation; Flow control; Navier Stokes equations; Optimal control; Quasi Newton algorithm

Indexed keywords

EID: 0000852499     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1064827595294678     Document Type: Article

References (65)

1. F. ABERGEL AND R. TEMAM, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynamics, 1 (1990), pp. 303-326.
2. G. V. ALEKSEYEV AND V. V. MALIKIN, Numerical analysis of optimal boundary control problems for the stationary Navier-Stokes equations, Comput. Fluid Dynamics J., 3 (1994), pp. 1-26.
3. E. ANDERSON, Z. BAI, C. BISCHOF, J. DEMMEL, J. DONGARRA, J. D. CROZ, A. GREENBAUM, S. HAMMARLING, A. MCKENNEY, S. OSTROUCHOV, AND D. SORENSEN, LAPACK Users' Guide, SIAM, Philadelphia, PA, 1992.
4. M. BERGGREN, Control and Simulation of Advection-diffusion Problems, Master's thesis, Department of Mechanical Engineering, University of Houston, Houston, TX, 1992.
5. M. BERGGREN, Optimal Control of Time Evolution Systems: Controllability Investigations and Numerical Algorithms, Ph.D. thesis, Department of Computational and Applied Mathematics, Rice University, Houston, TX, 1995.
6. M. BERGGREN, R. GLOWINSKI, AND J. L. LIONS, A computational approach to controllability issues for flow-related models. (I): Pointwise control of the viscous Burgers equation, Internat. J. Comput. Fluid Dynamics, 7 (1996), pp. 237-252.
7. M. BERGGREN, R. GLOWINSKI, AND J. L. LIONS, A computational approach to controllability issues for flow-related models. (II): Control of two-dimensional, linear advection-diffusion and Stokes models, Internat. J. Comput. Fluid Dynamics, 6 (1996), pp. 253-274.
8. C. BÖRGERS, Domain imbedding methods for the Stokes equations, Numer. Math., 57 (1990), pp. 73-187.
9. S. C. BRENNER AND L. R. SCOTT, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.
10. M. O. BRISTEAU, R. GLOWINSKI, AND J. PERIAUX, Numerical methods for the Navier-Stokes equations. Applications to the simulation of compressible and incompressible viscous flows, Comput. Phys. Reports, 6 (1987), pp. 435-451.
11. J. A. BURNS AND S. KANG, A Control Problem for Burgers' Equation with Bounded Input/Output, Tech. Report 90-45, ICASE, NASA Langley Research Center, Hampton, VA, 1990.
12. J. A. BURNS AND S. KANG, A stabilization problem for Burgers' equation with unbounded control and observation, in Estimation and Control of Distributed Parameter Systems: Proc. Internat. Conference on Control and Estimation of Distributed Parameter Systems, Vorau, July 8-14, 1990, Internat. Ser. Numer. Math., 100, Birkhauser-Verlag, Basel, Switzerland, 1991.
13. D. M. BUSHNELL AND C. B. MCGINLEY, Turbulence control in wall flows, Annual Rev. Fluid Mechanics, 21 (1989), pp. 1-20.
14. J. CAHOUET AND J. P. CHABARD, Some fast 3D solvers for the generalized Stokes problem, Internat. J. Numer. Methods Fluids, 8 (1988), pp. 869-895.
15. A. CAPATINA AND R. STAVRE, Numerical analysis of a control problem in heat conducting Navier-Stokes fluid, Internat. J. Engrg. Sci., 34 (1996), pp. 1467-1476.
16. A. CAPATINA AND R. STAVRE, Optimal control of a nonisothermal Navier-Stokes flow, Internat. J. Engrg. Sci., 34 (1996), pp. 59-66.
17. E. CASAS, The Navier-Stokes equations coupled with the heat equation: Analysis and control. Modelling, identification, sensitivity analysis and control of structures, Control Cybernet., 23 (1994), pp. 605-620.
18. G. CHAVENT, Identification of distributed parameter systems: About the output least square method, its implementation, and identifiability, in Proc. 5th IFAC Symposium on Identification and System Parameter Estimation, Vol. 1, R. Iserman, ed., Pergamon Press, Elmsford, NY, 1979, pp. 85-97.
19. H. CHOI, R. TEMAM, P. MOIN, AND J. KIM, Feedback control for unsteady flow and its application to the stochastic Burgers equation, J. Fluid Mech., 253 (1993), pp. 509-543.
20. P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
21. R. DAUTRAY AND J. L. LIONS, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 6: Evolution Problems II, Springer, Berlin, 1993.
22. E. DEAN, R. GLOWINSKI, AND C. H. Li, Supercomputer solutions of partial differential equation problems in computational fluid dynamics and control, Comput. Phys. Comm., 53 (1989), pp. 401-439.
23. E. J. DEAN AND P. GUBERNATIS, Pointwise control of Burgers' equation - a numerical approach, Comput. Math. Appl., 22 (1991), pp. 93-100.
24. J. E. DENNIS, M. HEINKENSCHLOSS, AND L. N. VICENTE, Trust-region Interior-point SQP Algorithms for a Class of Nonlinear Programming Problems, Tech. Report TR94-45, Department of Computational and Applied Mathematics, Rice University, Houston, TX, 1994.
25. J. E. DENNIS AND R. B. SCHNABEL, Numerical Methods for Unconstrained Optimization and Non-linear Equations, Prentice-Hall, Englewood Cliffs, NJ, 1983.
26. M. DESAI AND K. ITO, Optimal controls of Navier-Stokes equations, SIAM J. Control Optim., 32 (1994), pp. 1428-1446.
27. Q. Du, M. D. GUNZBURGER, AND L. S. Hou, Analysis and finite element approximation of optimal control problems for a Ladyzhenskaya model for stationary, incompressible, viscous flows, J. Comput. Appl. Math., 61 (1995), pp. 323-343.
28. H. O. FATTORINI AND S. S. SRITHARAN, Optimal chattering controls for viscous flow, Nonlinear Anal., 25 (1995), pp. 763-797.
29. O. GHATTAS AND J.-H. BARK, Optimal control of two- and three-dimensional Navier-Stokes flow, J. Comput. Phys, 163 (1997), pp. 231-244.
30. V. GIRAULT AND P.-A. RAVIART, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986.
31. R. GLOWINSKI, Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, New York, 1984.
32. R. GLOWINSKI, Finite Element Methods for the Numerical Solution of Incompressible Viscous Flow. Introduction to the Control of the Navier-Stokes Equations, in Lectures in Appl. Math. 28, American Mathematical Society, Providence, RI, 1991, pp. 219-301.
33. R. GLOWINSKI, Ensuring well-posedness through analogy: Stokes problem and boundary control of the wave equation, J. Comput. Phys., 104 (1993), pp. 189-221.
34. R. GLOWINSKI AND J. L. LIONS, Exact and approximate controllability for distributed parameter systems (Part II), in Acta Numerica, Cambridge Univ. Press, Cambridge 1995, pp. 159-333.
35. R. GLOWINSKI, T.-W. PAN, AND J. PERIAUX, A fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 112 (1994), pp. 133-148.
36. R. GLOWINSKI AND O. PIRONNEAU, Finite element methods for Navier-Stokes equations, Annual Rev. Fluid Mech., 24 (1992), pp. 167-204.
37. A. GRIEWANK, Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation, Optimization Methods and Software, 1 (1992), pp. 35-54.
38. M. D. GUNZBURGER, Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms, Academic Press, Boston, MA, 1989.
39. M. D. GUNZBURGER, L.-S. Hou, AND T. P. SVOBODNY, Numerical approximation of an optimal control problem associated with the Navier-Stokes equations, Appl. Math. Lett., 2 (1989), pp. 29-31.
40. M. D. GUNZBURGER, L.-S. Hou, AND T. P. SVOBODNY, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls, Math. Comp., 57 (1991), pp. 123-151.
41. M. D. GUNZBURGER, L.-S. Hou, AND T. P. SVOBODNY, Boundary velocity control of incompressible flow with an application to viscous drag reduction, SIAM J. Control Optim., 30 (1992), pp. 167-181.
42. M. D. GUNZBURGER ED., Flow Control, IMA Vol. Math. Appl. 68, Springer-Verlag, New York, 1995.
43. M. HEINKENSCHLOSS, Formulation and Analysis of a Sequential Quadratic Programming Method for the Optimal Boundary Control of Navier-Stokes Flow, Tech. Report TR97-14, Department of Computational and Applied Mathematics, Rice University, Houston, TX, 1997.
44. L. S. Hou, S. S. RAVINDRAN, AND Y. YAN, Numerical solution of optimal distributed control problems for incompressible flows, Internat. J. Comput. Fluid Dynamics, 8 (1997), pp. 99-114.
45. L. S. Hou AND Y. YAN, Dynamics for controlled Navier-Stokes systems with distributed controls, SIAM J. Control Optim., 35 (1997), pp. 654-677.
46. K. ITO AND S. S. RAVINDRAN, Optimal Control of Thermally Convected Fluid Flows, Tech. Report CSRC-TR96-8, Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC, 1996.
47. K. ITO AND S. S. RAVINDRAN, A Reduced Order Method for Simulation and Control of Fluid Flow, Tech. Report CSRC-TR96-27, Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC, 1996.
48. K. ITO AND S. S. RAVINDRAN, A Reduced Basis Method for Control Problems Governed by PDEs, Tech. Report CSRC-TR97-1, Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC, 1997.
49. A. JAMESON, Computational aerodynamics for aircraft design, Science, 245 (1989), pp. 361-371.
50. A. JAMESON, Optimum aerodynamic design using control theory, in Computational Fluid Dynamics Review, John Wiley, New York, 1995.
51. R. D. JOSLIN, M. D. GUNZBURGER, R. A. NICOLAIDES, G. ERLEBACHER, AND M. Y. HUSSAINI, Self-contained, Automated Methodology for Optimal Flow Control Validated for Transition Delay, AIAA J., 35 (1997), pp. 816-824.
52. M. KAWAHARA AND Y. SHIMADA, Gradient method of optimal control applied to the operation of a dam water gate, Internat. J. Numer. Methods Fluids, 19 (1994), pp. 463-477.
53. A. J. KEARSLEY, The Use of Optimization Techniques in the Solution of Partial Differential Equations from Science and Engineering, Ph.D. thesis, Department of Computational and Applied Mathematics, Rice University, Houston, TX, 1995.
54. F.-S. KUPFER AND E. W. SACHS, A prospective look at SQP methods for semilinear parabolic control problems, in Optimal Control of Partial Differential Equations: Proc. IFIP WG 7.2 International Conference, Irsee, April 9-12, 1990, Lecture Notes in Control and Inform. Sci. 143, K.-H. Hoffman and W. Krabs, eds., Springer-Verlag, Berlin, 1991, pp. 143-157.
55. F.-S. KUPFER AND E. W. SACHS, Numerical solution of a nonlinear parabolic control problem by a reduced SQP method, Comput. Optim. Appl., 1 (1992), pp. 113-135.
56. R. W. METCALFE, Boundary layer control: A brief review, in Computational Fluid Dynamics '94, Invited Lectures of the Second European CFD Conference, Stuttgart, Germany, S. Wagner, J. Periaux, and E. Hirschel, eds., John Wiley, New York, 1994, pp. 52-60.
57. J. NOCEDAL, Updating quasi-Newton matrices with limited storage, Math. Comp., 35 (1980), pp. 773-782.
58. O. PIRONNEAU, Finite Element Methods for Fluids, John Wiley, Chichester, U.K., 1989.
59. H. SCHLICHTING, Boundary-Layer Theory, McGraw Hill, New York, 1987.
60. S. S. SRITHARAN, Dynamic programming of Navier-Stokes equations, Systems Control Lett., 16 (1991), pp. 299-307.
61. S. S. SRITHARAN, An optimal control problem for exterior hydrodynamics, in Distributed Parameter Control Systems: New Trends and Applications, G. Chen, E. Lee, W. Littman, and L. Markus, eds., Marcel Dekker, New York, 1991, pp. 385-417.
62. W. W. SYMES AND J. O. BLANCH, Private communication, Department of Computational and Applied Mathematics, Rice University, Houston, TX, 1994.
63. R. TEMAM, Navier-Stokes Equations and Nonlinear Functional Analysis, SIAM, Philadelphia, PA, 1983.
64. A. S. W. THOMAS, Active wave control of boundary-layer transition, in Viscous Drag Reduction in Boundary Layers, D. M. Bushnell and J. N. Hefner, eds., Progress in Astronautics and Aeronautics 123, American Institute of Aeronautics and Astronautics, Washington, DC, 1990.
65. T. UMETSU, Y. TANAKA, AND M. KAWAHARA, Optimal control of flood using finite element method, Proc. JSCE, 446/1-19 (1992), pp. 23-32.