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

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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
Strony
413--430
Opis fizyczny
Bibliogr. 29 poz., rys., wykr.
Twórcy
  • Faculty of Mathematics, University of Łódz, ul. Banacha 22, 90–232 Łódź, Poland
autor
  • Institut Elie Cartan, Laboratoire de Mathématiques, Université Henri Poincaré Nancy I, B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex, France
autor
  • Institut Elie Cartan, Laboratoire de Mathématiques, Université Henri Poincaré Nancy I, B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex, France
  • Institut Elie Cartan, Laboratoire de Mathématiques, Université Henri Poincaré Nancy I, B.P. 239, 54506 Vandoeuvre-lès-Nancy Cedex, France
Bibliografia
  • [1] Allaire G., De Gournay F., Jouve F. and Toader A.M. (2005): Structural optimization using topological and shape sensitivity via a level set method. Control and Cybernetics, Vol. 34, No. 1, pp. 59-80.
  • [2] Amstutz S. and Andrä H. (2006): A new algorithm for topology optimization using a level-set method. - Journal of Computer Physics, Vol. 216, No. 2, pp. 573-588.
  • [3] Delfour M.C. and Zolesio J.-P. (2001): Shapes and Geometries. Philadelphia, PA: SIAM.
  • [4] Henrot A. and Pierre M. (2005): Variation et optimisation de formes: Une analyse géométrique. Berlin: Springer.
  • [5] Jackowska L., Sokołowski J., ˙Zochowski A. and Henrot A. (2002): On numerical solution of shape inverse problems. - Computational Optimization and Applications, Vol. 23, No. 2, pp. 231-255.
  • [6] Jackowska A.L., Sokołowski J. and ˙Zochowski A. (2003): Topological optimization and inverse problems. Computer Assisted Mechanics and Engineering Sciences, Vol. 10, No. 2, pp. 163-176.
  • [7] 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.
  • [8] Laurain A. (2006): Singularly perturbed domains in shape optimization. - Ph.D. thesis, Université de Nancy.
  • [9] 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.
  • [10] Maz'ya V., Nazarov S.A. and Plamenevskij B. (2000): Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vols. 1 and 2, Basel: Birkhäuser, p. 435.
  • [11] Nazarov S.A. (1999): Asymptotic conditions at a point, self adjoint extensions of operators, and the method of matched asymptotic expansions. American Mathematical Society Translations, Vol. 198, No. 2, pp. 77-125.
  • [12] 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.
  • [13] 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.
  • [14] Nazarov S.A. and Sokołowski J. (2004a): Self adjoint extensions for elasticity system in application to shape optimization. Bulletin of the Polish Academy of Sciences, Mathematics, Vol. 52, No. 3, pp. 237-248.
  • [15] Nazarov S.A. and Sokołowski J. (2004b): The topological derivative of the Dirichlet integral due to formation of a thin ligament. Siberian Mathematical Journal, Vol. 45, No. 2, pp. 341-355.
  • [16] 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 Universitatis Lodziensis, Vol. 12, pp. 39-72.
  • [17] Osher S. and Fedkiw R. (2004): Level Set Methods and Dynamic Implicit Surfaces. New York: Springer.
  • [18] 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.
  • [19] 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.
  • [20] 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.
  • [21] Sethian J. (1996): Level Set Methods. Cambridge: Cambridge University Press.
  • [22] Sokołowski J. and Zolesio J.-P. (1992): Introduction to shape optimization. Series in Computationnal Mathematics, Berlin: Springer Verlag, Vol. 16.
  • [23] Sokołowski J. and ˙Zochowski A. (1999): On the topological derivative in shape optimization. SIAMJournal on Control and Optimization, Vol. 37, No. 4, pp. 1251-1272.
  • [24] Sokołowski J. and ˙Zochowski A. (2001): Topological derivatives of shape functionals for elasticity systems. Mechanics of Structures and Machines, Vol. 29, No. 3, pp. 333-351.
  • [25] Sokołowski J. and ˙Zochowski A. (2003): Optimality conditions for simultaneous topology and shape optimization. SIAMJournal on Control and Optimization, Vol. 42, No. 4, pp. 1198-1221.
  • [26] Sokołowski J. and ˙Zochowski A. (2005a): Topological derivatives for contact problems. Numerische Mathematik, Vol. 102, No. 1, pp. 145-179.
  • [27] Sokołowski J. and ˙Zochowski A. (2005b): Topological derivatives for obstacle problems. Les prépublications de l'Institut Élie Cartan No. 12.
  • [28] Watson G.N. (1944): Theory of Bessel Functions. Cambridge: The University Press.
  • [29] 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
Identyfikator YADDA
bwmeta1.element.baztech-article-BPZ1-0041-0041
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