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Mathematical challenges in shape optimization

Treść / Zawartość
Identyfikatory
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Activities of the CNRS programme GDR shape optimization are described. Recent developments in shape optimization for eigenvalues and drag minimization are presented.
Rocznik
Strony
37--57
Opis fizyczny
Bibliogr. 59 poz., wykr.
Twórcy
autor
  • Ecole des Mines de Nancy et Institut Elie Cartan Nancy
  • Institut Elie Cartan Nancy
Bibliografia
  • Allaire, G. (2002) Shape Optimization by the Homogenization Method. Applied Mathematical Sciences 146. Springer-Verlag, New York.
  • Allaire, G. and Henrot, A. (2001) On some recent advances in shape optimization. C. R. Acad. Sci. Paris Ser. IIb Mecanique 329, 383-396.
  • Ammari, H. and Khelifi, A. (2003) Electromagnetic scattering by small dielectric inhomogeneities. J. Math. Pures Appl. 82 (7), 749–842.
  • Ammari, H. and Kang, H. (2004) Reconstruction of conductivity inhomogeneities of small diameter via boundary measurements. Inverse problems and spectral theory. Contemp. Math., 348, Amer. Math. Soc., Providence, RI, 23–32.
  • Ashbaugh, M.S. (1999) Open problems on eigenvalues of the Laplacian. In: T. M. Rassias and H. M. Srivastava, eds., Analytic and Geometric Inequalities and Their Applications, 4787, Kluwer.
  • Banichuk, N. (1990) Introduction to Optimization of Structures. Springer-Verlag, New York.
  • Bendsoe, M. (1995) Methods for Optimization of Structural Topology, Shape and Material. Springer Verlag.
  • Bendsoe, M. and Sigmund, O. (2003) Topology Optimization, Theory, Methods and Applications. Springer Verlag.
  • Bucur, D. and Buttazzo, G. (2005) Variational Methods in some Shape Optimization Problems. Lecture Notes of courses at Dipartimento di Matematica Universita di Pisa and Scuola Normale Superiore di Pisa, Series “Appunti di Corsi della Scuola Normale Superiore” and in Progress in Nonlinear Differential Equations and Their Applications. Birkhauser, Basel.
  • Bucur, D. and Henrot, A. (2000) Minimization of the third eigenvalue of the Dirichlet Laplacian. Proc. Roy. Soc. London 456, 985-996.
  • Buttazzo, G. and Dal Maso, G. (1993) An Existence Result for a Class of Shape Optimization Problems. Arch. Rational Mech. Anal. 122, 183-195.
  • Cagnol, J., Lasiecka, I., Lebiedzik, C. and Zolesio, J.P. (2002) Uniform stability in structural acoustic models with flexible curved walls. J. Differential Equations 186 (1), 88–121.
  • Cagnol, J. and Lebiedzik, C. (2004) On the free boundary conditions for a dynamic shell model based on intrinsic differential geometry. Appl. Anal. 83 (6), 607–633.
  • Canuto, B. (2002) Unique localization of unknown boundaries in a conducting medium from boundary measurements. ESAIM Control Optim. Calc. Var. 7, 1–22.
  • Chaabane, S., Elhechmi, C. and Jaoua, M. (2004) A stable recovery method for the Robin inverse problem. Math. Comput. Simulation 66 (4-5), 367–383.
  • Chambolle, A. and Larsen, C. (2003) C∞ regularity of the free boundary for a two-dimensional optimal compliance problem. Calc. Var. Partial Differential Equations 18 (1), 77–94.
  • Chenais, D. and Zuazua, E. (2003) Controllability of an elliptic equation and its finite difference approximation by the shape of the domain. Numer. Math. 95 (1), 63–99.
  • Choulli, M. (2003) Local stability estimate for an inverse conductivity problem. Inverse Problems 19 (4), 895–907.
  • Colton, D., Haddar, H. and Piana, M. (2003) The linear sampling method in inverse electromagnetic scattering theory. Special section on imaging. Inverse Problems 19 (6), S105–S137.
  • Comte, M. and Lachand-Robert, T. (2002) Functions and domains having minimal resistance under a single-impact assumption. SIAM J. Math. Anal. 34 (1), 101–120.
  • Delfour, M. and Zol´esio, J.P. (2001) Shapes and geometries. Analysis, differential calculus, and optimization. Advances in Design and Control SIAM, Philadelphia, PA.
  • Destuynder, P. (2001) Structures intelligentes pour le controle des bruits dans une tuyauterie. C. R. Acad. Sci. Paris Ser. I Math. 333 (10), 961–966.
  • El Badia, A. and Ha-Duong, T. (2002) On an inverse source problem for the heat equation. Application to a pollution detection problem. J. Inverse Ill-Posed Probl. 10 (6), 585–599.
  • Faber, G. (1923) Beweis, dass unter allen homogenenMembranen von gleicher Flache und gleicher Spannung die kreisformige den tiefsten Grundton gibt. Sitz. Ber. Bayer. Akad. Wiss., 169-172.
  • Feireisl, G.E. (2003) Shape optimisation in viscous compressible fluids. Appl. Math. Optim. 47, 59-78.
  • Feireisl, G.E. (2004) Dynamics of Viscous Compressible Fluids. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press.
  • Habbal, A. (2002) Generation of optimal periodic oscillations for the control of boundary layers. SIAM Journal on Control and Optimization 41 (3), 712-722.
  • Haslinger, J. and Makinen, R.A.E. (2003) Introduction to Shape Optimization. Theory, Approximation, and Computation. Advances in Design and Control SIAM, Philadelphia, PA.
  • Haslinger, J. and Neittaanmaki, P. (1996) Finite Element Approximation for Optimal Shape, Material and Topology Design, Second edition. John Wiley & Sons, Ltd., Chichester.
  • Haug, E.J. and Rousselet, B. (1980) Design sensitivity analysis in structural mechanics. II. Eigenvalue variations. J. Structural Mech. 8 (2), 161-186.
  • Hebrard, P. and Henrot, A. (2003) Optimal Shape and Position of the Actuators for the Stabilization of a String. Optimization and control of distributed systems. Systems Control Lett. 48 (3-4), 199–209.
  • Henrot, A. (2003) Minimization problems for eigenvalues of the Laplacian. Journal of Evolution Equations, special issue dedicated to Philippe Benilan, 3, 443-461.
  • Henrot, A. and Oudet, E. (2003) Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions. Arch. Ration. Mech. Anal. 169 (1), 73-87.
  • Henrot, A. and Pierre, M. (2005) Variation and optimisation de forme. Mathematiques et Applications. Springer, Berlin.
  • Hild, P., Ionescu, I.R., Lachand-Robert, T. and Rosca, I. (2002) The blocking of an inhomogeneous Bingham fluid. Applications to landslides. M2AN Math. Model. Numer. Anal. 36 (6), 1013–1026.
  • Ha-Duong, T., Jaoua, M. and Menif, F. (2004) A modified frozen Newton method to identify a cavity by means of boundary measurements. Math. Comput. Simulation 66 (4-5), 355–366.
  • Kawohl, B., Pironneau, O., Tartar, L. and Zolesio, J.P. (2000) Optimal Shape Design. Lectures given at the Joint C.I.M./C.I.M.E. Summer School held in Troia, June 1–6, 1998. Edited by A. Cellina and A. Ornelas. Springer Lecture Notes in Mathematics, 1740.
  • Krahn, E.(1924) Uber eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann. 94, 97-100.
  • Krahn, E. (1926) Uber Minimaleigenschaften der Kugel in drei un mehr Dimensionen. Acta Comm. Univ. Dorpat. A9, 1-44.
  • Lachand-Robert, T. and Peletier, M.A. (2001) Newtons problem of the body of minimal resistance in the class of convex developable functions. Math. Nachr. 226, 153–176.
  • Laporte, E. and Le Tallec, P. (2003) Numerical Methods in Sensitivity Analysis and Shape Optimization. Birkhauser.
  • Lions, P. L. (1998) Mathematical Topics in Fluid Mechanics. V. 2 Compressible Models. Clarendon Press.
  • Marco, N., Lanteri, S., Desideri, J.-A. and Periaux, J. (1999) A Parallel Genetic Algorithm for Multi-Objective Optimization in Computational Fluid Dynamics. In: K. Miettinen et al., eds., Evolutionary Algorithms in Engineering and Computer Science, John Wiley.
  • Mazya, V. G., Nazarov, S. A. and Plamenevskii, B. A. (2000) Asymptotics of Solutions to Elliptic Boundary-Value Problems Under a Singular Perturbation of the Domain. Tbilisi: Tbilisi Univ. 1981 (Russian). Asymptotische Theorie elliptischer Randwertaufgaben in singular gestorten Gebieten. 1, 2. Berlin: Akademie-Verlag. 1991. Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains Vol. 1, 2, Basel: Birkhauser Verlag.
  • Mohammadi, B. and Pironneau, O. (2001) Applied Shape Optimization for Fluids. Clarendon Press, Oxford.
  • Nazarov, S.A. and Plamenevsky, B.A. (1994) Elliptic Problems in Domains with Piecewise Smooth Boundaries. De Gruyter Exposition in Mathematics 13, Walter de Gruyter.
  • Nazarov, S.A. and Sokołowski, J. (2003) Asymptotic analysis of shape functionals. Journal de Mathematiques pures et appliquees, 82 (2), 125-196.
  • Oudet, E. (2004) Some numerical results about minimization problem involving eigenvalues. ESAIM COCV 10, 315-335.
  • Plotnikov, P.I. and Sokołowski, J. (2002) On compactness, domain dependence and existence of steady state solutions to compressible isothermal Navier-Stokes equations. Les Prepublications de l' Institut Elie Cartan 35, Nancy, 1-37. Also Journal of Mathematical Fluid Mechanics 7 (2005), in press. RR-5156: http://www.inria.fr/rrrt/rr-5156.html.
  • Plotnikov, P.I. and Sokołowski, J. (2004) Stationary boundary value problems for Navier-Stokes equations with adiabatic index. Doklady Mathematics 70 (1), 535-538, translated from Doklady Akademii Nauk, 397, 1-6.
  • Plotnikov, P.I. and Sokołowski, J. (2004) Concentrations of stationary solutions to compressible Navier-Stokes equations. Les Prepublications de l’ Institut Elie Cartan 15, Nancy; also to appear in Communications in Mathematical Physics; RR-5481: http://www.inria.fr/rrrt/rr-5481.html.
  • Pironneau, O. (1984) Optimal Shape Design for Elliptic Systems. Springer-Verlag, New York.
  • Polya, G. (1955) On the characteristic frequencies of a symmetric membrane. Math. Z. 63, 331–337.
  • Rousselet, B. (1983) Shape design sensitivity of a membrane. J. Optim. Theory Appl. 40 (4), 95-623.
  • Rousselet, B. (2002) Sensitivity of dynamic structures, case of a smart beam. J. Convex Anal. 9 (2), 649–663.
  • Samet, B., Amstutz, S. and Masmoudi, M. (2003) The topological asymptotic for the Helmholtz equation. SIAM J. Control Optim. 42 (5), 1523–1544.
  • Simon, J. (1980) Differentiation with respect to the domain in boundary value problems. Num. Funct. Anal. Optimiz. 2, 649-687.
  • Sokołowski, J. and Zolesio, J. P. (1992) Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer Series in Computational Mathematics 10, Springer, Berlin.
  • Wolf, S.A. and Keller, J.B. (1994) Range of the first two eigenvalues of the Laplacian. Proc. R. Soc. London A 447, 397-412.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BAT5-0007-0080
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