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Boundary integral representations of second derivatives in shape optimization

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Abstrakty
EN
For a shape optimization problem second derivatives are investigated, obtained by a special approach for the description of the boundary variation and the use of a potential ansatz for the state. The natural embedding of the problem in a Banach space allows the application of a standard differential calculus in order to get second derivatives by a straight forward "repetition of differentiation". Moreover, by using boundary value characerizations for more regular data, a complete boundary integral representation of the second derivative of the objective is possible. Basing on this, one easily obtains that the second derivative contains only normal components for stationary domains, i.e. for domains, satisfying the first order necessary condition for a free optimum. Moreover, the nature of the second derivative is discussed, which is helpful for the investigation of sufficient optimality conditions.
Twórcy
  • Technical University of Chemnitz, Faculty of Mathematics, D-09107 Chemnitz, Germany
Bibliografia
  • [1] S. Belov and N. Fujii, Symmetry and sufficient condition of optimality in a domain optimization problem, Control and Cybernetics 26 (1) (1997), 45-56.
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  • [3] K. Eppler, Optimal shape design for elliptic equations via BIE-methods, preprint TU Chemnitz, 1998.
  • [4] K. Eppler, Optimal shape design for elliptic equations via BIE-methods, submitted (1999).
  • [5] K. Eppler, Fréchet-differentiability and sufficient optimality conditions for shape functionals, in: K.-H. Hoffmann, G. Leugering, F. Tröltzsch (eds.), Optimal control of Partial Differential Equations, ISNM, 133 (1999), 133-143, Basel: Birkhäuser.
  • [6] K. Eppler, Boundary integral representations of second derivatives in shape optimization, dfg-preprint series 'Echtzeit-Optimierung grosser Systeme' preprint 00-1 (2000), http://www.tu-chemnitz.de/eppler
  • [7] K. Eppler, Second derivatives and sufficient optimality conditions for shape functionals, to appear in Control and Cybernetics (2000).
  • [8] N. Fujii, Second order necessary conditions in a domain optimization problem, Journal of Optimization Theory and Applications 65 (2) (1990), 223-245.
  • [9] N. Fujii, Sufficient conditions for optimality in shape optimization, Control and Cybernetics 23 (3) (1994), 393-406.
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  • [15] A.M. Khludnev and J. Sokolowski, Modelling and Control in Solid Mechanics, Birkhäuser, Basel 1997.
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  • [19] J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization, Springer, Berlin 1992.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1005
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