Inverse shape optimization problems and application to airfoils

• Autorzy: Desideri J.-A. , Zolesio J.-P.
• Języki publikacji: EN

Abstrakty

EN

We consider a set of parameterized planar arcs (x(t), y(t)) (0<t<l), satisfying certain smoothness, regularity and monotonicity conditions (in particuler x(t) is monotone increasing, and y(t) positive and unimodal), and a functional J(y) involving an adjustable weighting function omega(t) and a positive constant o > 1. We first prove the strict convexity of the functional for alpha > 2. Under the less stringent condition alpha > 1, we derive the stationarity condition and the formal expression for the Hessian, and prove that if a point exists at which the functional is stationary w.r.t. variations in y = y(t), for fixed x = x(t), then it is unique and realizes a global minimum; the functional is then unimodal. We also observe that the stationarity condition (Euler-Lagrange quation) is an integral-differential equation depending only on the arc shape and not on the parameterization per se, which gives the variational problem a certain intrinsic character. Then, we solve the inverse problem: given an admissible parameterized arc, we construct a smooth weighting function omega(t) for which the stationarity condition is satisfied, thus making the functional unimodal, and derive certain asymptotics. A numerical example pertaining to optimum-shape design in aerodynamics is computed for illustration.

Słowa kluczowe

EN

Partial-Differential Equations (PDEs); computational methods; shape optimization; calculus of variations

PL

równanie różniczkowe cząstkowe; metody obliczeniowe; optymalizacja kształtu; rachunek wariacyjny

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2005
• Tom: Vol. 34, no 1
• Strony: 165--202
• Opis fizyczny: Bibliogr. 9 poz., wykr.

Twórcy

autor

Desideri J.-A.

• INRIA, OPALE Project 2004 Route des Lucioles, BP 93, F–06902 Sophia Antipolis Cedex, France

autor

Zolesio J.-P.

• INRIA, OPALE Project 2004 Route des Lucioles, BP 93, F–06902 Sophia Antipolis Cedex, France

Bibliografia

• Bélahcène, F. and Désidéri, J.-A. (2003) Paramétrisation de Bézier adaptative pour l’optimisation de forme en Aérodynamique. Research Report 4943, INRIA.
• Clarich, A. and Désidéri, J.-A. (2002) Self-adaptive parameterisation for aerodynamic optimum-shape design. Research Report 4428, INRIA.
• Désidéri, J.-A. (2003) Hierarchical optimum-shape algorithms using embedded Bézier parameterizations. In: Yu. Kuznetsov, P. Neittanmäki and O. Pironneau, eds., Numerical Methods for Scientific Computing, Variational Problems and Applications. CIMNE, Barcelona.
• Farin, G. (1990) Curves and Surfaces for Computer Aided Geometric Design – A practical Guide author. W. Rheinboldt and D. Siewiorek, eds., Computer Science and Scientific Computing, Academic Press, 2nd, Boston.
• Jabri, Y. (2003) The Mountain Pass Theorem, Variants, Generalizations and Some Applications. Cambridge University Press.
• Karakasis, M. and Désidéri, J.-A. (2002) Model Reduction and Adaption of Optimum-Shape Design in Aerodynamics by Neural Networks. Research Report 4503, INRIA.
• Périaux, J., Bugeda, G., Chaviaropoulos, P.K., Giannakoglou, K., Lantéri, S. and Mantel, B. (1998) Optimum Aerodynamic Design & Parallel Navier-Stokes Computations, ECARP European Computational Aerodynamics Research Project. Notes on Numerical Fluid Mechanics, Vieweg, Braunschweig/Wiesbaden, Germany.
• Sederberg, T.W. (2004) Computer Aided Geometric Design. CS557 class notes, http://cagd.cs.byu.edu/∼tom/courses.html Brigham Young University, Utah.
• Tang, Z.L. and Désidéri, J.-A. (2002) Towards Self-Adaptive Parameterization of Bézier Curves for Airfoil Aerodynamic Design. Research Report 4572, INRIA.