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2007 | 17 | 3 | 413-430
Tytuł artykułu

A level set method in shape and topology optimization for variational inequalities

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The level set method is used for shape optimization of the energy functional for the Signorini problem. The boundary variations technique is used in order to derive the shape gradients of the energy functional. The conical differentiability of solutions with respect to the boundary variations is exploited. The topology modifications during the optimization process are identified by means of an asymptotic analysis. The topological derivatives of the energy shape functional are employed for the topology variations in the form of small holes. The derivation of topological derivatives is performed within the framework proposed in (Sokołowski and Żochowski, 2003). Numerical results confirm that the method is efficient and gives better results compared with the classical shape optimization techniques.
Rocznik
Tom
17
Numer
3
Strony
413-430
Opis fizyczny
Daty
wydano
2007
poprawiono
2006-05-10
(nieznana)
2006-12-15
Twórcy
  • Faculty of Mathematics, University of Łódź, ul. Banacha 22, 90-232 Łódź, Poland
  • Institut Elie Cartan, Laboratoire de Mathématiques, Université Henri Poincaré Nancy I, B.P. 239, 54506 Vandoeuvre-les-Nancy Cedex, France
  • Institut Elie Cartan, Laboratoire de Mathématiques, Université Henri Poincaré Nancy I, B.P. 239, 54506 Vandoeuvre-les-Nancy Cedex, France
  • Institut Elie Cartan, Laboratoire de Mathématiques, Université Henri Poincaré Nancy I, B.P. 239, 54506 Vandoeuvre-les-Nancy Cedex, France
Bibliografia
  • Allaire G., De Gournay F., Jouve F. and Toader A.M. (2005): Structural optimizationusing topological and shape sensitivity via a level set method. Control and Cybernetics, Vol.34, No.1, pp.59-80.
  • Amstutz S. and Andra H. (2006): A new algorithm for topology optimization using a level-set method. - Journal of Computer Physics, Vol.216, No.2, pp.573-588.
  • Delfour M.C. and Zolesio J.-P. (2001): Shapes and Geometries. Philadelphia, PA: SIAM.
  • Henrot A. and Pierre M. (2005): Variation et optimisation de formes: Une analyse geometrique. Berlin: Springer.
  • Jackowska L., Sokołowski J., Żochowski A. and Henrot A. (2002): On numerical solution of shape inverse problems. - Computational Optimization and Applications, Vol.23, No.2, pp.231-255.
  • Jackowska A.L., Sokołowski J. and Żochowski A. (2003): Topological optimization and inverse problems. Computer Assisted Mechanics and Engineering Sciences, Vol.10, No.2, pp.163-176.
  • Jarusek J., Krbec M., Rao M. and Sokołowski J. (2003): Conical differentiability for evolution variational inequalities. Journal of Differential Equations, Vol.193, No.1, pp.131-146.
  • Laurain A. (2006): Singularly perturbed domains in shape optimization. - Ph.D. thesis, Université de Nancy.
  • Masmoudi M. (2002):The topological asymptotic, In: Computational Methods for Control Applications (R.Glowinski, H.Kawarada and J.Periaux, Eds.). GAKUTO Int. Ser. Math. Sci. Appl., Vol.16, pp.53-72.
  • Mazya V., Nazarov S.A. and Plamenevskij B. (2000): Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vols.1 and 2, Basel: Birkhauser, p.435.
  • Nazarov S.A. (1999): Asymptotic conditions at a point, self adjoint extensions of operators, and the method of matched asymptotic expansions. American Mathematical Society Tranations, Vol.198, No.2, pp.77-125.
  • Nazarov S.A. and Sokołowski J. (2003a): Self adjoint extensions of differential operators in application to shape optimization. Comptes Rendus Mécanique, Vol.331, No.10, pp.667-672.
  • Nazarov S.A. and Sokołowski J. (2003b): Asymptotic analysis of shape functionals. Journal de Mathématiques pures et appliquées, Vol.82, No.2, pp.125-196.
  • Nazarov S.A. and Sokołowski J. (2003c): Asymptotic analysis of shape functionals. Journal of de Mathématiques pures et appliquées, Vol.82, No.2, pp.125-196.
  • Nazarov S.A. and Sokołowski J. (2004a): Self adjoint extensions for elasticy system in application to shape optimization. Bulletin of the Polish Academy of Sciences, Mathematics, Vol.52, No.3, pp. 237-248.
  • Nazarov S.A. and Sokołowski J. (2004b): The topological derivative of the Dirichlet integral due to formation of a thinligament. Siberian Mathematical Journal, Vol.45, No.2, pp.341-355.
  • Nazarov S.A., Slutskij A.S. and Sokołowski J. (2005): Topological derivative of the energy functional due to formation of a thin ligament on a spatial body. Folia Mathematicae, Acta Universatis Lodziensis, Vol.12, pp.39-72.
  • Osher S. and Fedkiw R. (2004): Level Set Methods and Dynamic Implic Surfaces. New York: Springer.
  • Osher S. and Sethian J. (1988): Fronts propagating with curvature-dependant speed: Algorithms based on Hamilton-Jacobi formulation. Journal of Computational Physics, Vol.79, No.1, pp.12-49.
  • Peng D., Merriman B., Osher S., Zhao S. and Kang M. (1999): A PDE-based fast local level set method. Journal of Computational Physics, Vol.155, No.2, pp.410-438.
  • Rao M. and Sokołowski J. (2000): Tangent sets in Banach spaces and applications to variational inequalities. Les prépublications de l'Institut Élie Cartan, No.42.
  • Sethian J. (1996): Level Set Methods. Cambridge: Cambridge University Press.
  • Sokołowski J. and Zolesio J.-P. (1992): Introduction to shape optimization. Series in Computationnal Mathematics, Berlin: Springer Verlag, Vol.16.
  • Sokołowski J. and Żochowski A. (1999): On the topological derivative in shape optimization. SIAM Journal on Control and Optimization, Vol.37, No.4, pp.1251-1272.
  • Sokołowski J. and Żochowski A. (2001): Topological derivatives of shape functionals for elasticy systems. Mechanics of Structures and Machines, Vol.29, No.3, pp.333-351.
  • Sokołowski J. and Żochowski A. (2003): Optimaly conditions for simultaneous topology and shape optimization. SIAM Journal on Control and Optimization, Vol.42, No.4, pp.1198-1221.
  • Sokołowski J. and Żochowski A. (2005a): Topological derivatives for contact problems. Numerische Mathematik, Vol.102, No.1, pp.145-179.
  • Sokołowski J. and Żochowski A. (2005b): Topological derivatives for obstacle problems. Les prépublications de l'Institut Élie Cartan No.12.
  • Watson G.N. (1944):Theory of Bessel Functions. Cambridge: The University Press.
  • Zhao H.K., Chan T., Merriman B. and Osher S. (1996): A variational level set approach to multi-phase motion. Journal of Computational Physics, Vol.127, No.1, pp.179-195
Typ dokumentu
Bibliografia
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