×

A recovered gradient method applied to smooth optimal shape problems. (English) Zbl 0870.65050

The article is devoted to sensitivity analysis in the optimal shape design. Several state problems based on elliptic boundary value problems are considered. The problems are formulated on a “generalized rectangle” whose upper side is formed by a design function which is a Bézier curve. The paper presents a new postprocessing technique suitable for quasiuniform triangulations which can be employed in the sensitivity analysis of optimal shape design problems. In the first part of the paper, a class of admissible domains is introduced, and the state problems and cost functionals are defined. Then, sensitivity formulae expressed by boundary integrals are derived for three standard cost functionals. There is also a section where the discretization by linear finite elements is discussed and the approximate optimization problems are defined. Finally, the authors recall the technique of recovered gradients on chevron triangulations and propose its application to the sensitivity formulae evaluation.

MSC:

65K10 Numerical optimization and variational techniques
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
90C52 Methods of reduced gradient type
PDFBibTeX XMLCite
Full Text: EuDML

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. · Zbl 0682.65003
[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. · Zbl 0323.90063 · doi:10.1007/BF01447854
[3] C. de Boor: A Practical Guide to Splines. Springer-Verlag, New York, 1978. · 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. · 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. · Zbl 0831.65071
[7] P.G. Ciarlet: Basic error estimates for elliptic problems, Handbook of Numerical Analysis II (P.G. Ciarlet, J.L. Lions. North-Holland, Amsterdam, 1991.
[8] J. Haslinger, P. Neittaanmäki: Finite Element Approximation for Optimal Shape Design, Theory and Applications. John Wiley, Chichester, 1988. · Zbl 0713.73062
[9] E.J. Haug, K.K. Choi and V. Komkov: Design Sensitivity Analysis of Structural Systems. Academic Press, Orlando, London, 1986. · Zbl 0618.73106
[10] I. Hlaváček: Optimization of the domain in elliptic problems by the dual finite element method. Apl. Mat. 30 (1985), 50-72. · Zbl 0575.65103
[11] I. Hlaváček, R. Mäkinen: On the numerical solution of axisymmetric domain optimization problems. Appl. Math. 36 (1991), 284-304. · Zbl 0745.65044
[12] I. Hlaváček, M. Křížek and Pištora: How to recover the gradient of linear elements on nonuniform triangulations. Appl. Math. 41 (1996), 241-267. · Zbl 0870.65093
[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. · Zbl 0861.65091
[14] I. Hlaváček: Shape optimization by means of the penalty method with extrapolation. Appl. Math 39 (1994), 449-477. · Zbl 0826.65056
[15] J.T. King, S.M. Serbin: Boundary flux estimates for elliptic problems by the perturbed variational method. Computing 16 (1976), 339-347. · Zbl 0338.65054 · doi:10.1007/BF02252082
[16] M. Křížek, P. Neittaanmäki: On superconvergence techniques. Acta Appl. Math. 9 (1987), 175-198. · Zbl 0624.65107 · doi:10.1007/BF00047538
[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. · Zbl 0797.65079 · doi:10.1007/BF01385871
[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. · Zbl 0737.73094
[19] N. Levine: Superconvergent recovery of the gradient from piecewise linear finite element approximations. IMA J. Numer. Anal. 5 (1985), 407-427. · Zbl 0584.65067 · doi:10.1093/imanum/5.4.407
[20] P.A. Raviart, J.M. Thomas: Primal hybrid finite element method for 2nd order elliptic equations. Math. Comp. 31 (1977), 391-413. · Zbl 0364.65082 · doi:10.2307/2006423
[21] J. Sokolowski, J.P. Zolesio: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer-Verlag, Berlin, 1992. · Zbl 0487.49004
[22] L.B. Wahlbin: Superconvergence in Galerkin finite element methods (Lecture notes). Cornell University 1994, 1-243.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.