##
**Shape optimization in two-dimensional elasticity by the dual finite element method.**
*(English)*
Zbl 0611.73021

The paper deals with a shape optimization of a part of the boundary of a two-dimensional elastic body according to minimum of a cost functional of stresses. An approximate cost functional of stresses is evaluated directly by means of a piecewise linear stress field. Zero displacements prescribed on the part of the boundary is taken as the design variable.

A convergence of approximations has been studied for two optimization problems, the cost functional of which represents: i) a generalization of the well-known von Mises or Tresca criterion, ii) a suitable norm of the reaction forces on the unknown part of the boundary. Two optimization problems have been formulated in terms of stresses. The existence of an optimal solution has been proved. Finite element approximations of the dual state problem were introduced and a continuous dependence of the approximate stress functions on the approximate control has been proved. The existence of a subsequence of approximate controls, which converges to an optimal control function has been proved.

The paper has a cognizable character, but it is far from technical applications.

A convergence of approximations has been studied for two optimization problems, the cost functional of which represents: i) a generalization of the well-known von Mises or Tresca criterion, ii) a suitable norm of the reaction forces on the unknown part of the boundary. Two optimization problems have been formulated in terms of stresses. The existence of an optimal solution has been proved. Finite element approximations of the dual state problem were introduced and a continuous dependence of the approximate stress functions on the approximate control has been proved. The existence of a subsequence of approximate controls, which converges to an optimal control function has been proved.

The paper has a cognizable character, but it is far from technical applications.

Reviewer: St.Jendo

### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |

49J40 | Variational inequalities |

74P99 | Optimization problems in solid mechanics |

### Keywords:

minimization of cost functional; respect to part of boundary; elastic body fixed; existence of optimal boundary; Castigliano principle; approximate cost functional of stresses; piecewise linear stress field; convergence; Finite element approximations; dual state problem; continuous dependence; approximate stress functions; approximate control
PDF
BibTeX
XML
Cite

\textit{I. Hlaváĉek}, RAIRO, Modélisation Math. Anal. Numér. 21, 63--92 (1987; Zbl 0611.73021)

### References:

[1] | J. P. AUBIN, Approximations of elliptic boundary value problems. J. Wiley-Interscience, New York, 1972. Zbl0248.65063 MR478662 · Zbl 0248.65063 |

[2] | N. V. BANICHUK, Problems and methods of optimal structural design. PlenumN. V. BANICHUK, Problems and me Press, New York and London, 1983. Zbl0649.73041 MR715778 · Zbl 0649.73041 |

[3] | D. BEGIS, R. GLOWINSKI, Application de la méthode des éléments finis à l’approximation d’un problème de domaine optimal. Appl. Math. & Optim. 2 (1975), 130-169. Zbl0323.90063 MR443372 · Zbl 0323.90063 |

[4] | [4] I. HLAVACEK, Convergence of an equilibrium finite element model for plane elastostatics. Apl. Mat.24 (1979), 427-456. Zbl0441.73101 MR547046 · Zbl 0441.73101 |

[5] | I. HLAVACEK, Dual finite element analysis for some elliptic variational equations and inequalities. Acta Applicandae Math. 1 (1983), 121-150. Zbl0523.65049 MR713475 · Zbl 0523.65049 |

[6] | [6] J. HLAVACEK : Optimization of the domain in elliptic problems by the dual finite element method. Api.Mat.30 (1985), 50-72. Zbl0575.65103 MR779332 · Zbl 0575.65103 |

[7] | J. HASLINGER, I. HLAVACEK, Approximation of the Signorini problem with friction by a mixed finite element method. J. Math. Anal. Appl. 86 (1982), 99-122. Zbl0486.73099 MR649858 · Zbl 0486.73099 |

[8] | J. HASLINGER, J. LOVISEK, The approximation of the optimal shape control problem governed by a variational inequality with flux cost functional. To appear in Proc. MR831811 · Zbl 0498.73069 |

[9] | J. HASLINGER, P. NEITTAANMÄKI, On the existence of optimal shapes in contact problems, Numer. Funct. Anal, and Optimiz. 7 (1984), 107-124. Zbl0559.73099 MR767377 · Zbl 0559.73099 |

[10] | J. HASLINGER, P. NEITTAANMÄKI, T TIIHONEN, On optimal shape design of an elastic body on a rigid foundation. To appear in Proc. of the MAFELAP Confe-rence 1984. Zbl0588.73159 MR811062 · Zbl 0588.73159 |

[11] | J. NECAS, I. HLAVACEK, Mathematical theory of elastic and elasto-plastic bodies.Elsevier, Amsterdam 1981. Zbl0448.73009 · Zbl 0448.73009 |

[12] | V. B WATWOOD, B. J. HARTZ, An equilibrium stress field model for finite element solution of two-dimensional elastostatic problems. Internat. J. Solids Structures 4 (1968), 857-873. Zbl0164.26201 · Zbl 0164.26201 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.