We study the controllability of linearized shape-dependent operators for flow problems. The first operator is a mapping from the shape of the computational domain to the tangential wall velocity of the potential flow problem and the second operator maps to the wall shear stress of the Stokes problem. We derive linearizations of these operators, provide their well-posedness and finally show approximate controllability. The controllability of the linearization shows in what directions the observable can be changed by applying infinitesimal shape deformations.
Accepté le :
DOI : 10.1051/cocv/2016012
Mots clés : Controllablility, shape-dependent operator, shape optimization, shape derivative, partial differential equation, inverse problem
@article{COCV_2017__23_3_751_0,
author = {Leith\"auser, C. and Pinnau, R. and Fe{\ss}ler, R.},
title = {Approximate controllability of linearized shape-dependent operators for flow problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {751--771},
publisher = {EDP-Sciences},
volume = {23},
number = {3},
year = {2017},
doi = {10.1051/cocv/2016012},
mrnumber = {3660447},
zbl = {1365.93044},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2016012/}
}
TY - JOUR AU - Leithäuser, C. AU - Pinnau, R. AU - Feßler, R. TI - Approximate controllability of linearized shape-dependent operators for flow problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 751 EP - 771 VL - 23 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016012/ DO - 10.1051/cocv/2016012 LA - en ID - COCV_2017__23_3_751_0 ER -
%0 Journal Article %A Leithäuser, C. %A Pinnau, R. %A Feßler, R. %T Approximate controllability of linearized shape-dependent operators for flow problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 751-771 %V 23 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016012/ %R 10.1051/cocv/2016012 %G en %F COCV_2017__23_3_751_0
Leithäuser, C.; Pinnau, R.; Feßler, R. Approximate controllability of linearized shape-dependent operators for flow problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 751-771. doi : 10.1051/cocv/2016012. http://archive.numdam.org/articles/10.1051/cocv/2016012/
, and , Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194 (2004) 363–393. | DOI | MR | Zbl
H. Amann and J. Escher, Analysis II. Birkhäuser (2008). | MR
J. Anderson and J. Wendt, Vol. 206 of Computational fluid dynamics, McGraw-Hill (1995). | Zbl
and , Controllability of an elliptic equation and its finite difference approximation by the shape of the domain. Numer. Math. 95 (2003) 63–99. | DOI | MR | Zbl
M. Delfour and J. Zolésio, Vol. 22 of Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization. Society for Industrial Mathematics (2010). | MR | Zbl
R. Eppler, Airfoil design and data. Springer Berlin (1990).
, Analytical and numerical methods in shape optimization. Math. Methods Appl. Sci. 31 (2008) 2095–2114. | DOI | MR | Zbl
W. Hess and S. Ulbrich, An inexact penalty SQP algorithm for PDE-constrained optimization with an application to shape optimization in linear elasticity. Optim. Methods Softw. (2012). | MR | Zbl
C. Leithäuser, Shape Design for Stokes Flows. Diplomarbeit, TU Kaiserslautern (2009).
C. Leithäuser, Controllability of Shape-dependent Operators and Constrained Shape Optimization for Polymer Distributors. Ph. D. thesis, TU Kaiserslautern (2013).
C. Leithäuser and R. Feßler, Characterizing the image space of a shape-dependent operator for a potential flow problem. Appl. Math. Lett. (2012). | MR | Zbl
J. Marburger, Space-Mapping and Optimal Shape Design. Diplomarbeit, TU Kaiserslautern (2007).
B. Mohammadi and O. Pironneau, Applied shape optimization for fluids. Oxford University Press, USA (2001). | MR | Zbl
and , Shape optimization in fluid mechanics. Ann. Rev. Fluid Mech. 36 (2004) 255–279. | DOI | MR | Zbl
and , Boundary controllability of a stationary stokes system with linear convection observed on an interior curve. J. Optim. Theory Appl. 99 (1998) 201–234. | DOI | MR | Zbl
and , On the controllability of the Laplace equation observed on an interior curve. Rev. Mat. Complut. 11 (1998) 403–441. | DOI | MR | Zbl
, and , A phase-field model for compliance shape optimization in nonlinear elasticity. ESAIM: COCV 18 (2012) 229–258. | Numdam | MR | Zbl
O. Pironneau, Optimal shape design for elliptic systems. Springer (1984). | MR | Zbl
and , Optimal control and shape optimization of aorto-coronaric bypass anastomoses. Math. Models Methods Appl. Sci. 13 (2003) 1801–1824. | DOI | MR | Zbl
, On optimization, control and shape design of an arterial bypass. Int. J. Numer. Methods Fluids 47 (2005) 1411–1419. | DOI | MR | Zbl
, Differentiation with respect to the domain in boundary value problems. Numer. Funct. Anal. Optim. 2 (1980) 649–687. | DOI | MR | Zbl
J. Sokolowski and J. Zolesio, Vol. 16 of Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer-Verlag (1992). | MR | Zbl
, , and , New possibilities with sobolev active contours. Int. J. Comput. Vision 84 (2009) 113–129. | DOI | Zbl
J. Wloka, Partial differential equations. Cambridge University Press (1987). | MR | Zbl
Cité par Sources :
