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Title: Shape Hessian for generalized Oseen flow by differentiability of a minimax: A Lagrangian approach (English)
Author: Gao, Zhiming
Author: Ma, Yichen
Author: Zhuang, Hongwei
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 57
Issue: 3
Year: 2007
Pages: 987-1011
Summary lang: English
Category: math
Summary: The goal of this paper is to compute the shape Hessian for a generalized Oseen problem with nonhomogeneous Dirichlet boundary condition by the velocity method. The incompressibility will be treated by penalty approach. The structure of the shape gradient and shape Hessian with respect to the shape of the variable domain for a given cost functional are established by an application of the Lagrangian method with function space embedding technique. (English)
Keyword: shape sensitivity analysis
Keyword: shape Hessian
Keyword: Eulerian semiderivative
Keyword: differentiability of a minimax
Keyword: Oseen flow
MSC: 49K35
MSC: 49Q12
MSC: 76D07
MSC: 76D55
idZBL: Zbl 1174.76008
idMR: MR2356935
Date available: 2009-09-24T11:51:08Z
Last updated: 2020-07-03
Stable URL:
Reference: [1] R. A. Adams: Sobolev Spaces.Academic Press, London, 1975. Zbl 0314.46030, MR 0450957
Reference: [2] J. Céa: Lectures on Optimization: Theory and Algorithms.Springer-Verlag, 1978. MR 0545791
Reference: [3] J. Céa: Problems of shape optimal design.“Optimization of Distributed Parameter Structures”, Vol. II, E. J. Haug and J. Céa (eds.), Sijthoff and Noordhoff, Alphen aan denRijn, Netherlands, 1981, pp. 1005–1048.
Reference: [4] R. Correa and A. Seeger: Directional derivative of a minmax function.Nonlinear Analysis, Theory Methods and Applications 9 (1985), 13–22. MR 0776359, 10.1016/0362-546X(85)90049-5
Reference: [5] M. C. Delfour, G. Payre and J. -P. Zolésio: An optimal triangulation for second order elliptic problems.Computer Methods in Applied Mechanics and Engineering 50 (1985), 231–261. MR 0800331, 10.1016/0045-7825(85)90095-7
Reference: [6] M. C. Delfour and J.-P. Zolésio: Shape sensitivity analysis via min max differentiability.SIAM J. Control and Optimization 26 (1988), 834–862. MR 0948649, 10.1137/0326048
Reference: [7] M. C. Delfour and J.-P. Zolésio: Computation of the shape Hessian by a Lagrangian method.IFAC, Control of Distributed Parameter Systems. Perpignan, France, 1989, pp. 215–220.
Reference: [8] M. C. Delfour and J.-P. Zolésio: Shape Hessian by the velocity method: a Lagrangian approach.Lect. Notes. Cont. Inf. Sci, 147, Springer-Verlag, 1990, pp. 255–279. MR 1179447
Reference: [9] M. C. Delfour and J.-P. Zolésio: Tangential calculus and shape derivative in Shape Optimization and Optimal Design (Cambridge, 1999), Dekker, New York.2001, pp. 37–60. MR 1812358
Reference: [10] M. C. Delfour and J.-P. Zolésio: Shapes and Geometries: Analysis, Differential Calculus, and Optimization.Advance in Design and Control, SIAM (2002). MR 1855817
Reference: [11] I. Ekeland and R. Temam: Convex Analysis and Variational Problems.Series Classics in Applied Mathematics, SIAM, Philadelphia, 1999. MR 1727362
Reference: [12] N. Fujii: Domain optimization problems with a boundary value problem as a constraint.“Control of Distributed Parameter Systems” 1986, Pergamon Press, Oxford, New York, 1986, pp. 5–9.
Reference: [13] D. Gilbarg and N. S. Trudinger: Elliptic Partial Differential Equations of Second Order.Springer, Berlin, 1983. MR 0737190
Reference: [14] J. Hadamard: Mémoire sur le problème d’analyse relatif à l’équilibre des plaques élastiques encastrées.Mém. Sav. étrang. (2) 33, Nr. 4, 128 S (1908).
Reference: [15] J. Haslinger and R. A. E. Mäkinen: Introduction to Shape Optimization: Theory, Approximation, and Computation.SIAM (2003). MR 1969772
Reference: [16] J.-C. Nédélec: Acoustic and Electromagnetic Equations. Integral representations for harmonic problems.Springer-Verlag, Berlin Heidelberg New York, 2001. MR 1822275
Reference: [17] O. Pironneau: Optimal Shape Design for Elliptic systems.Springer, Berlin, 1984. Zbl 0534.49001, MR 0725856
Reference: [18] J. Simon: Differentiation with respect to the domain in boundary value problems.Numer. Funct. Anal. Optim. 2 (1980), 649–687. Zbl 0471.35077, MR 0619172, 10.1080/01630563.1980.10120631
Reference: [19] J. Simon: Second variations for domain optimization problems.“Control of Distributed Parameter Systems”, Birkhauser Verlag, 1988. MR 1033071
Reference: [20] R. Temam: Navier-Stokes Equations: Theory and Numerical Analysis.Providence, RI: American Mathematical Society (AMS), 2001. Zbl 0981.35001, MR 1846644
Reference: [21] J.-P. Zolésio: Identification de domaines par déformation, Thèse de doctorat d’état, Université de Nice, France.1979.
Reference: [22] J. Sokolowski and J.-P. Zolésio: Introduction to Shape Optimization: Shape Sensitivity Analysis.Springer-Verlag, Berlin, 1992. MR 1215733


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