Analysis of the Hessian for aerodynamic optimization: Inviscid flow. (English) Zbl 0969.76015
Summary: We analyze inviscid aerodynamic shape optimization problems governed by the full potential and Euler equations in two and three dimensions. The analysis indicates that minimization of pressure-dependent cost functions results in Hessians whose eigenvalue distributions are identical for the full potential and Euler equations. However, the optimization problems in two and three dimensions are inherently different. While the two-dimensional optimization problems are well-posed, the three-dimensional ones are ill-posed. Oscillations in the shape up to the smallest scale allowed by the design space can develop in the direction perpendicular to the flow, implying that a regularization is required. The analysis also gives an estimate of the Hessian’s condition number which implies that the problems at hand are ill-conditioned. Infinite-dimensional approximations for the Hessians are constructed, and preconditioners for gradient-based methods are derived from these approximate Hessians. Numerical results are given for small-disturbance potential equation in two and three space dimensions.
MSC:
| 76B75 | Flow control and optimization for incompressible inviscid fluids |
| 65K10 | Numerical optimization and variational techniques |
Keywords:
full potential equation; two-dimensional problem; three-dimensional problem; inviscid aerodynamic shape optimization; Euler equations; pressure-dependent cost functions; Hessians; eigenvalue distributions; regularization; preconditioners; gradient-based methods; small-disturbance potential equationReferences:
| [1] | Pironneau, O., Optimal shape design for elliptic systems. Optimal shape design for elliptic systems, Springer series in computational physics (1983), Springer-Verlag: Springer-Verlag Berlin · Zbl 0496.93029 |
| [2] | Mantel B, Periaux J, Stoufflet B. Optimum design methods in aerodynamics, AGARD-FDP-VKI Special Course, April 25-29, 1994; Mantel B, Periaux J, Stoufflet B. Optimum design methods in aerodynamics, AGARD-FDP-VKI Special Course, April 25-29, 1994 |
| [3] | Jameson, A., Aerodynamic design via control theory, J. Sci. Comput, 3, 233-260 (1988) · Zbl 0676.76055 |
| [4] | Iollo A, Salas MD. Contribution to the optimal shape design of two-dimensional internal flows with embedded shocks, ICASE Report No. 95-20, 1995; Iollo A, Salas MD. Contribution to the optimal shape design of two-dimensional internal flows with embedded shocks, ICASE Report No. 95-20, 1995 |
| [5] | Ta’asan, S.; Kuruvila, G.; Salas, M. D., Aerodynamic design and optimization in one shot, (AIAA 92-0025, 30th Aerospace Sciences Meeting & Exhibit (1992)) |
| [6] | Ta’asan, S., Trends in aerodynamics design and optimization: a mathematical viewpoint, (AIAA 95-1730, 12th AIAA Computational Fluid Dynamics Conference (1995)) |
| [7] | Dervieux, A.; Malé, J.; Macro, N.; Périaux, J.; Stoufflet, B.; Chen, H. Q., Some recent advances in optimal shape design for aeronautical flows, (Proceedings of ECCOMAS, 2nd Computational Fluid Dynamics Conference (1994)) |
| [8] | Frank, P. D.; Shubin, G. R., A comparison of optimization-based approaches for a model computational aerodynamics design problem, J. Comput. Phys, 98, 74 (1992) · Zbl 0741.76067 |
| [9] | Jou, W. H.; Huffman, W. P.; Young, D. P.; Melvin, R. G.; Bieterman, M. B.; Hilmes, C. L.; Johnson, F. T., Practical considerations in aerodynamic design optimization, (AIAA 95-1730, 12th AIAA Computational Fluid Dynamics Conference (1995)), (The Boeing Company) · Zbl 0923.76234 |
| [10] | Hadamard, J., Lecon sur le calcul des variation (1910), Gautheir-Villadrs: Gautheir-Villadrs Paris · JFM 41.0432.02 |
| [11] | Lighthill MJ. A new method of two-dimensional aerodynamic design. Aeronautical Research Council, R & M 1111, 1945; Lighthill MJ. A new method of two-dimensional aerodynamic design. Aeronautical Research Council, R & M 1111, 1945 |
| [12] | Gill, P. E.; Murray, W.; Wright, M. H., Practical optimization (1981), Academic Press: Academic Press New York · Zbl 0503.90062 |
| [13] | Arian E. Multigrid methods for optimal shape design governed by elliptic systems. Ph.D. Thesis, The Weizmann Institute of Science, Israel, 1994; Arian E. Multigrid methods for optimal shape design governed by elliptic systems. Ph.D. Thesis, The Weizmann Institute of Science, Israel, 1994 |
| [14] | Arian E, Ta’asan S. Multigrid one shot methods for optimal design problems: infinite-dimensional control. ICASE Report No. 94-52, 1994; Arian E, Ta’asan S. Multigrid one shot methods for optimal design problems: infinite-dimensional control. ICASE Report No. 94-52, 1994 |
| [15] | Arian, E.; Ta’asan, S., (Borggaard, J.; Burkardt, J.; Gunzburger, M.; Peterson, J., Shape optimization in one shot: optimal design and control (1995), Birkhäuser: Birkhäuser Boston) · Zbl 0826.76076 |
| [16] | Arian, E.; Ta’asan, S., Smoothers for optimization problems, (Proceedings of the Seventh Copper Mountain Conference on Multigrid Methods (1995)) |
| [17] | Hirsch, C., (Numerical computation of internal and external flows, vol. 2 (1988), John Wiley and Sons: John Wiley and Sons New York) · Zbl 0662.76001 |
| [18] | Arian, E., On the coupling of aerodynamic and structural design, J. Comput. Phys, 135, 83-96 (1997) · Zbl 0888.76036 |
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.
