Shape Derivative of Drag Functional
SIAM Journal on Control and Optimization
Shape optimization of an airfoil in the presence of compressible and viscous flows
Computational Optimization and Applications
| Hi-index | 0.00 |
The minimization of the drag functional for the stationary, isentropic, compressible Navier-Stokes equations (NSE) in three spatial dimensions is considered. In order to establish the existence of an optimal shape, the general result on compactness of families of generalized solutions to the NSE is established within in the framework of the modern theory of nonlinear PDEs [P. L. Lions, Mathematical Topics in Fluid Dynamics. Vol. $2$. Compressible Models, Oxford University Press, Clarendon Press, New York, 1998], [E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, Oxford, UK, 2004]. The family of generalized solutions to the NSE is constructed over a family of admissible domains $\mathcal{U}_{ad}$. Any admissible domain $\Omega=B\setminus S$ contains an obstacle $S$, e.g., a wing profile. Compactness properties of the family of admissible domains in the form of the condition ($\mathcal{H}_\Omega$) is imposed. Roughly speaking, the condition ($\mathcal{H}_\Omega$) is satisfied, provided that for any sequence of admissible domains $\{\Omega_n\}\subset \mathcal{U}_{ad}$ there is a subsequence convergent both in Hausdorff metrics and in the sense of Kuratowski and Mosco. The analysis is performed for the adiabatic constant $\gamma1$ in the pressure law $p(\rho)=\rho^\gamma$ and it is based on the technique used in [P. I. Plotnikov and J. Sokolowski, Comm. Math. Phys., 258 (2005), pp. 567-608] in the case of discretized NSE. The result is a generalization to the stationary equations with $\gamma1$ of the results obtained in [E. Feireisl, A. H. Novotny´, and H. Petzeltova´, Math. Methods Appl. Sci., 25 (2002), pp. 1045-1073], [E. Feireisl, Appl. Math. Optim., 47 (2003), pp. 59-78] for evolution equations within the range $\gamma3/2$ for the adiabatic ratio.