Optimization of the domain in elliptic problems by the dual finite element method

Ivan Hlaváček

Aplikace matematiky (1985)

  • Volume: 30, Issue: 1, page 50-72
  • ISSN: 0862-7940

Abstract

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An optimal part of the boundary of a plane domain for the Poisson equation with mixed boundary conditions is to be found. The cost functional is (i) the internal energy, (ii) the norm of the external flux through the unknown boundary. For the numerical solution of the state problem a dual variational formulation - in terms of the gradient of the solution - and spaces of divergence-free piecewise linear finite elements are used. The existence of an optimal domain and some convergence results are proved.

How to cite

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Hlaváček, Ivan. "Optimization of the domain in elliptic problems by the dual finite element method." Aplikace matematiky 30.1 (1985): 50-72. <http://eudml.org/doc/15384>.

@article{Hlaváček1985,
abstract = {An optimal part of the boundary of a plane domain for the Poisson equation with mixed boundary conditions is to be found. The cost functional is (i) the internal energy, (ii) the norm of the external flux through the unknown boundary. For the numerical solution of the state problem a dual variational formulation - in terms of the gradient of the solution - and spaces of divergence-free piecewise linear finite elements are used. The existence of an optimal domain and some convergence results are proved.},
author = {Hlaváček, Ivan},
journal = {Aplikace matematiky},
keywords = {dual finite element method; optimal domain; Thomson principle; rate of convergence; numerical examples; dual finite element method; optimal domain; Thomson principle; rate of convergence; numerical examples},
language = {eng},
number = {1},
pages = {50-72},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Optimization of the domain in elliptic problems by the dual finite element method},
url = {http://eudml.org/doc/15384},
volume = {30},
year = {1985},
}

TY - JOUR
AU - Hlaváček, Ivan
TI - Optimization of the domain in elliptic problems by the dual finite element method
JO - Aplikace matematiky
PY - 1985
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 30
IS - 1
SP - 50
EP - 72
AB - An optimal part of the boundary of a plane domain for the Poisson equation with mixed boundary conditions is to be found. The cost functional is (i) the internal energy, (ii) the norm of the external flux through the unknown boundary. For the numerical solution of the state problem a dual variational formulation - in terms of the gradient of the solution - and spaces of divergence-free piecewise linear finite elements are used. The existence of an optimal domain and some convergence results are proved.
LA - eng
KW - dual finite element method; optimal domain; Thomson principle; rate of convergence; numerical examples; dual finite element method; optimal domain; Thomson principle; rate of convergence; numerical examples
UR - http://eudml.org/doc/15384
ER -

References

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  1. D. Begis R. Glowinski, 10.1007/BF01447854, Appl. Math. & Optim. 2 (1975), 130-169. (1975) MR0443372DOI10.1007/BF01447854
  2. J. Haslinger I. Hlaváček, Convergence of a finite element method based on the dual variational formulation, Apl. Mat. 21 (1976), 43 - 65. (1976) MR0398126
  3. I. Hlaváček, The density of solenoidal functions and the convergence of a dual finite element method, Apl. Mat. 25 (1980), 39-55. (1980) MR0554090
  4. I. Hlaváček, 10.1007/BF00046832, Acta Applic. Math. 1, (1983), 121 - 20. (1983) MR0713475DOI10.1007/BF00046832
  5. J. Haslinger J. Lovíšek, The approximation of the optimal shape problem governed by a variational inequality with flux cost functional, To appear in Proc. Roy. Soc. Edinburgh. 
  6. I. Hlaváček J. Nečas, Optimization of the domain in elliptic unilateral boundary value problems by finite element method, R.A.I.R.O. Anal. numér, 16, (1982), 351 - 373. (1982) MR0684830
  7. M. Křížek, Conforming equilibrium finite element methods for some elliptic plane problems, R.A.I.R.O. Anal. numér, 17, (1983), 35-65. (1983) MR0695451
  8. J. Haslinger P. Neittaanmäki, On optimal shape design of systems governed by mixed Dirichlet-Signorini boundary value problems, To appear in Math. Meth. Appl. Sci. MR0845923
  9. P. Neittaanmäki T. Tiihonen, Optimal shape design of systems governed by a unilateral boundary value problem, Lappeenranta Univ. of Tech., Dept. of Physics and Math., Res. Kept. 4/1982. (1982) 
  10. B. A. Murtagh, 10.1007/BF01588950, Math. Programming, 14 (1978), 41-72. (1978) Zbl0383.90074MR0462607DOI10.1007/BF01588950
  11. R. Fletcher, Practical methods of optimization, vol. 2, constrained optimization, J. Wiley, Chichester, 1981. (1981) Zbl0474.65043MR0633058

Citations in EuDML Documents

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  1. I. Hlaváček, Shape optimization in two-dimensional elasticity by the dual finite element method
  2. Ivan Hlaváček, Domain optimization in 3 D -axisymmetric elliptic problems by dual finite element method
  3. Jan Chleboun, Hybrid variational formulation of an elliptic state equation applied to an optimal shape problem
  4. Zdeněk Kestřánek, Numerical analysis for optimal shape design in elliptic boundary value problems
  5. Vladislav Pištora, Shape optimization of an elasto-plastic body for the model with strain- hardening
  6. Ivan Hlaváček, Jan Chleboun, A recovered gradient method applied to smooth optimal shape problems

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