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Keywords:
shape optimization; penalty method; extrapolation; finite elements
shape optimization; penalty method; extrapolation; finite elements
Summary:
A model shape optimal design in $\mathbb{R}^2$ is solved by means of the penalty method with extrapolation, which enables to obtain high order approximations of both the state function and the boundary flux, thus offering a reliable gradient for the sensitivity analysis. Convergence of the proposed method is proved for certain subsequences of approximate solutions.
References:
[1] I. Babuška: The finite element method with penalty. Math. Comp. 27 (1973), 221–228. DOI 10.1090/S0025-5718-1973-0351118-5 | MR 0351118
[2] I. Babuška: Numerical solution of partial differential equations. Preprint, March 1973, Univ. of Maryland. MR 0366062
[3] P.G. Ciarlet: Basic error estimates for elliptic problems. In: Handbook of Numer. Anal., vol. II, Finite element methods (Part 1), ed. by P.G. Ciarlet and J.L. Lions, Elsevier, (North-Holland), 1991. MR 1115237 | Zbl 0875.65086
[4] S. Conte, C. de Boor: Elementary numerical analysis; an algorithmic approach. McGraw-Hill, New York, 1972. MR 0202267
[5] P. Grisvard: Boundary value problems in non-smooth domains. Univ. of Maryland, Lecture Notes #19, 1980.
[6] P. Grisvard: Singularities in boundary value problems. RMA 22, Res. Notes in Appl. Math., Masson, Paris, Springer-Verlag, Berlin, 1992. MR 1173209 | Zbl 0778.93007
[7] E.J. Haug, K.K. Choi, V. Komkov: Design sensitivity analysis of structural systems. Academic Press, Orlando-London, 1986. MR 0860040
[8] I. Hlaváček: Penalty method and extrapolation for axisymmetric elliptic problems with Dirichlet boundary conditions. Apl. Mat. 35 (1990), 405–417. MR 1072609
[9] J. Chleboun, R. Mäkinen: Primal formulation of an elliptic equation in smooth optimal shape problems. Advances in Math. Sci. Appl.
[10] J. Kadlec: On the regularity of the solution of the Poisson problem on a domain with boundary locally similar to the boundary of a convex open set. Czechoslovak Math. J. 14 (1964), no. 89, 386–393.. MR 0170088 | Zbl 0166.37703
[11] J.T. King: New error bounds for the penalty method and extrapolation. Numer. Math. 23 (1974), 153–165. DOI 10.1007/BF01459948 | MR 0400742 | Zbl 0272.65092
[12] J.T. King, S.M. Serbin: Boundary flux estimates for elliptic problems by the perturbed variational method. Computing, 16 (1976), 339–347. DOI 10.1007/BF02252082 | MR 0418485
[13] J.T. King, S.M. Serbin: Computational experiments and techniques for the penalty method with extrapolation. Math. Comp. 32 (1978), 111–126. DOI 10.1090/S0025-5718-1978-0471866-0 | MR 0471866
[14] J.L. Lions, E. Magenes: Problèmes aux limites non homogènes et applications. vol. 1, Dunod, Paris, 1968. MR 0247243
[15] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. MR 0227584
[16] M. Zlámal: Curved elements in the finite element method I. SIAM J. Num. Anal. 10 (1973), 229–240. DOI 10.1137/0710022 | MR 0395263
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