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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.
Reviewer: St.Jendo

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
49J40 Variational inequalities
74P99 Optimization problems in solid mechanics
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References:

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