Abstract
Variational method is applied to the state equations in order to derive the costate equations and their boundary conditions. Thereafter, the analyses of the eigenvalues of the state and costate equations are performed. It is shown that the eigenvalues of the Jacobean matrices of the state and the transposed Jacobean matrices of the costate equations are analytically and numerically the same. Based on the eigenvalue analysis, the costate equations with their boundary conditions are numerically integrated. Numerical results of the eigenvalues problems of the state and costate equations and of a maximization problem are finally presented.
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References
Pironneau O (June 1973) On optimum profile in stokes flow. J Fluid Mech 59:117–128
Meric AR (1984) Boundary element methods for the optimization of distributed parameter systems. Int J Numer Meth Eng 20:1291–1304
Koda M (1984) Optimum design in fluid mechanical distributed-parameter systems. Large Scale Syst 6(3):279–291
Shubin RG, Frank DP (1991) A comparison of the implicit gradient approach and the variational approach to aerodynamic design optimization. Boeing Computer Services, WA
Jameson A (1988) Aerodynamic design via control theory. ICASE Report No 88–64, NASA Langley Research Center, Hampton, VA
Cabuk H, Modi V (1992) Optimum plane diffusers in laminar flow. J Fluid Mech 237:373–393
Ibrahim AH, Hou Gene JW, Tiwari SN, Smith RE (2002) Aerodynamic design optimization using Euler equations and variational methods. Comp Fluids 31:227–243
Gelfand IM, Fomin SV (1965) Calculus of variations. Prentice Hall, Englewood Cliffs, NJ
Hoffmann K, Chiang ST (1993) Computational fluid dynamics for engineers, vols I and II. Engineering Education System, Wichita, KS
Ibrahim AH, Tiwari SN, Smith RE (1999) Stability analysis and aerodynamical design optimization using variational methods. Int J Numer Meth Eng 44:1709–1726
Demuren AO, Ibraheem SO (1993) On the stability analysis of approximate factorization methods for 3D Euler and Navier-Stokes equations. NASA TM 106314, ICOMP-93–29
Acknowledgements
This work was supported through FAR Grant NAG-4-165 by the NASA Dryden Flight Research Center and through Cooperative Agreement NCC1–68 by the NASA Langley Research Center. Dr. K. K. Gupta was the technical monitor for the NASA Dryden NAG-4-165 grant.
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Ibrahim, A.H., Tiwari, S.N. A variational method in design optimization and sensitivity analysis for aerodynamic applications. Engineering with Computers 20, 88–95 (2004). https://doi.org/10.1007/s00366-004-0273-7
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DOI: https://doi.org/10.1007/s00366-004-0273-7