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Collaborator: J. Sprekels , D. Tiba
Cooperation with: V. Arnautu (``Alexandru Ioan Cuza'' University, Iasi, Romania)
Description: The importance, and the many technical applications, of various curved mechanical structures like arches, curved rods, shells are well known. The scientific literature concerning their modeling is very rich, see [1], [2]. We only mention the classes of polynomial models and the asymptotic models.
The aim of this research project is the study of optimization problems naturally associated to such models. In the previous paper [3], the Kirchhoff-Love model for arches and its optimization have been discussed. In [4], we concentrate on the case of polynomial models for curved rods and shells. The model introduced for shells is of a generalized Naghdi type ([5]); it involves a system of six partial differential equations of elliptic type in two independent variables.
The minimization parameter is the shape of the curved rod or of the shell. The thickness is supposed to remain constant, which holds in many applications. Such problems are known in the literature as shape optimization problems. As the geometry of the curved rod or of the shell is parametrized by certain functions, we obtain control-by-the-coefficients problems.
The results obtained concern existence of minimizers, sensitivity analysis, and numerical experiments.
The existence question is intimately connected to the coercivity inequalities for the elliptic operator, uniformly with respect to the geometry. In the case of shells, an inequality of Korn's type was proved which is valid for the class of all the parametrized shells from some ball.
For the optimality conditions, the directional derivatives of the performance indices with respect to variations in the geometry have been investigated. The variations used for curved rods preserve the length of the rod, which is an important constraint for possible applications. This enables to use gradient algorithms for numerical experiments. The computed examples are curved rods.
References:
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