Shape sensitivity analysis of the work functional for the compressible Navier–Stokes equations

Pavel I. Plotnikov (1) , Jan Sokolowski (2)

1 Lavryentyev Institute of Hydrodynamics
2 IECL - Institut Élie Cartan de Lorraine

The compressible Navier–Stokes equations with nonhomogeneous Dirichlet conditions in a bounded domain with an obstacle are considered (P.I. Plotnikov, J. Sokolowski, Compressible Navier-Stokes Equations. Theory and Shape Optimization, Birkhäuser, Basel, 2012). The dependence of local solutions on the shape of an obstacle is analyzed (P.I. Plotnikov, E.V. Ruban, J. Sokołowski, SIAM J. Math. Anal. 40:1152–1200, 2008; P.I. Plotnikov, E.V. Ruban, J. Sokołowski, J. Math. Pures Appl. 92:113–162, 2009; P.I. Plotnikov, J. Sokołowski, Dokl. Akad. Nauk 397:166–169, 2004; P.I. Plotnikov, J. Sokołowski, J. Math. Fluid Mech. 7:529–573, 2005; P.I. Plotnikov, J. Sokołowski, Comm. Math. Phys. 258:567–608, 2005; P.I. Plotnikov, J. Sokołowski, SIAM J. Control Optim. 45:1165–1197, 2006; P.I. Plotnikov, J. Sokołowski, Uspekhi Mat. Nauk 62:117–148, 2007; P.I. Plotnikov, J. Sokołowski, Stationary boundary value problems for compressible Navier-Stokes equations, in Handbook of Differential Equations: Stationary Partial Differential Equations, vol. VI, Elsevier/North-Holland, Amsterdam, 2008, pp. 313–410; P.I. Plotnikov, J. Sokołowski, SIAM J. Control Optim. 48:4680–4706, 2010; P.I. Plotnikov, J. Sokołowski, J. Math. Sci. 170:34–130, 2010). The shape derivatives (J. Sokołowski, J.-P. Zolésio, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992) of solutions to the compressible Navier–Stokes equations are derived. The shape gradient (J. Sokołowski, J.-P. Zolésio, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992) of the work functional is obtained. In this way the framework for numerical methods of shape optimization (P. Plotnikov, J. Sokołowski, A. Żochowski, Numerical experiments in drag minimization for compressible Navier-Stokes flows in bounded domains, in Proceedings of the 14th International IEEE/IFAC Conference on Methods and Models in Automation and Robotics, MMAR’09, 2009, 4 pp; A. Kaźmierczak, P.I. Plotnikov, J. Sokołowski, A. Żochowski, Numerical method for drag minimization in compressible flows, in 15th International Conference on Methods and Models in Automation and Robotics (MMAR), pp. 97–101, 2009.

Mots clés: Shape derivative; Rigid obstacle; Nonhomogeneous Dirichlet boundary value problem for nonstationary; Work functional; Shape functional; Shape gradient; Compressible Navier–Stokes equations; Drag functional

Domaines: Equations aux dérivées partielles [math.AP]

Pavel I. Plotnikov, Jan Sokolowski. Shape sensitivity analysis of the work functional for the compressible Navier–Stokes equations. Lecture Notes in Computational Science and Engineering, 2014, Optimization with PDE constaints, 101, pp.343-378. ⟨10.1007/978-3-319-08025-3_11⟩. ⟨hal-01096487⟩