Previous |  Up |  Next

Article

Keywords:
shape optimization; sensitivity analysis; superconvergence; recovered gradient.
Summary:
A new postprocessing technique suitable for nonuniform triangulations is employed in the sensitivity analysis of some model optimal shape design problems.
References:
[1] R.H. Bartels, J. C. Beatty and B.A. Barsky: An Introduction to Splines for use in Computer Graphics and Geometric Modelling. Morgan Kaufmann, Los Altos, 1987. MR 0919732
[2] D. Begis, R. Glowinski: Application de la méthode des éléments finis à l’approximation d’un probléme de domaine optimal. Appl. Math. Optim. 2 (1975), 130–169. MR 0443372
[3] C. de Boor: A Practical Guide to Splines. Springer-Verlag, New York, 1978. MR 0507062 | Zbl 0406.41003
[4] V. Braibant, C. Fleury: Aspects theoriques de l’optimisation de forme par variation de noeuds de controle, in Conception optimale de formes (Cours et Séminaires). Tome II, INRIA, Nice, 1983.
[5] J. Chleboun: Hybrid variational formulation of an elliptic state equation applied to an optimal shape problem. Kybernetika 29 (1993), 231–248. MR 1231869 | Zbl 0805.49024
[6] J. Chleboun, R.A.E. Mäkinen: Primal hybrid formulation of an elliptic equation in smooth optimal shape problems. Adv. in Math. Sci. and Appl. 5 (1995), 139–162. MR 1325963
[7] P.G. Ciarlet: Basic error estimates for elliptic problems, Handbook of Numerical Analysis II (P.G. Ciarlet, J.L. Lions eds.). North-Holland, Amsterdam, 1991. MR 1115237
[8] J. Haslinger, P. Neittaanmäki: Finite Element Approximation for Optimal Shape Design, Theory and Applications. John Wiley, Chichester, 1988. MR 0982710
[9] E.J. Haug, K.K. Choi and V. Komkov: Design Sensitivity Analysis of Structural Systems. Academic Press, Orlando, London, 1986. MR 0860040
[10] I. Hlaváček: Optimization of the domain in elliptic problems by the dual finite element method. Apl. Mat. 30 (1985), 50–72. MR 0779332
[11] I. Hlaváček, R. Mäkinen: On the numerical solution of axisymmetric domain optimization problems. Appl. Math. 36 (1991), 284–304.
[12] I. Hlaváček, M. Křížek and Pištora: How to recover the gradient of linear elements on nonuniform triangulations. Appl. Math. 41 (1996), 241–267. MR 1395685
[13] I. Hlaváček, M. Křížek: Optimal interior and local error estimates of a recovered gradient of linear elements on nonuniform triangulations. To appear in Journal of Computation. MR 1414854
[14] I. Hlaváček: Shape optimization by means of the penalty method with extrapolation. Appl. Math 39 (1994), 449–477. MR 1298733
[15] J.T. King, S.M. Serbin: Boundary flux estimates for elliptic problems by the perturbed variational method. Computing 16 (1976), 339–347. MR 0418485
[16] M. Křížek, P. Neittaanmäki: On superconvergence techniques. Acta Appl. Math. 9 (1987), 175–198. MR 0900263
[17] R.D. Lazarov, A.I. Pehlivanov, S.S. Chow and G.F. Carey: Superconvergence analysis of the approximate boundary flux calculations. Numer. Math. 63 (1992), 483–501. MR 1189533
[18] R.D. Lazarov, A.I. Pehlivanov: Local superconvergence analysis of the approximate boundary flux calculations. Proceed. of the Conference Equadiff 7, Teubner-Texte zur Math., Bd 118, Leipzig 1990, 275–278.
[19] N. Levine: Superconvergent recovery of the gradient from piecewise linear finite element approximations. IMA J. Numer. Anal. 5 (1985), 407–427. MR 0816065 | Zbl 0584.65067
[20] P.A. Raviart, J.M. Thomas: Primal hybrid finite element method for 2nd order elliptic equations. Math. Comp. 31 (1977), 391–413. MR 0431752
[21] J. Sokolowski, J.P. Zolesio: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer-Verlag, Berlin, 1992. MR 1215733
[22] L.B. Wahlbin: Superconvergence in Galerkin finite element methods (Lecture notes). Cornell University 1994, 1–243. MR 1439050
Partner of
EuDML logo