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Description:
The aim is to develop new approaches to the analysis and the optimization of curved mechanical structures like curved rods and shells . This is a continuation of the investigations on arches [1] and beams and plates [4]. Much attention is paid to the relaxation of the smoothness assumptions on the geometry of the mechanical structures.
(a) Control variational methods.
In the case of arches and plates, a new variational approach based on
optimal control theory allows minimal regularity assumptions : plates
with discontinuous thickness and absolutely continuous arches. In the
latter case, if the tangent vector is non-zero a.e., then the classical
reparametrization gives Lipschitzian representation of the arches
with unit tangent vectors. It should be noted that the classical
variational formulation of the Kirchhoff-Love model for arches
requires three times bounded derivability of the
parametrization. Moreover, the new method allows to prove the
continuity and the Gâteaux/Fréchet differentiability of the
mapping coefficient solution, in very weak norms. These results can be
found in the papers [1], [5].
We are using a model that has been developed and studied, both from a theoretical and a numerical point of view, in [2]. It consists of a system of nine ordinary differential equations with null boundary conditions (the rod is clamped). Besides the deformation of the line of centroids, the cross-section may change its shape as well. The class of admissible geometries for the optimum problem is obtained by first generating the unit tangent vectors
The examined problem consists of the minimization of a general
cost functional depending on the unknown geometry and the deformation
Continuity and Gâteaux differentiability with
respect to of the deformation
are proved
in
, equipped with the strong
topologies. This allows to establish the existence of optimal shapes
for problem (P) and the writing of the
first order optimality conditions. Numerical tests will be performed
as well, [7].
We consider the Naghdi-type shell model introduced
in [6]. It is assumed that the middle
surface of the shell is the graph of a function open, bounded, (multiply)
connected, and the thickness is constant.
The shape optimization problem is expressed as above, by the
minimization of a general cost functional depending on the geometry
and on the deformation
. Constraints of the type
, closed bounded subset, may be as well
imposed.
To study the continuity and the differentiability properties of the
mapping is very difficult, since Korn-type
inequalities with constants independent of the unknown geometry
are necessary. We have proved such inequalities by using an
extension method. In this way, we have established the existence of at
least one optimal geometry, when
is compact, and the first order optimality conditions
[8]. It should be noted that the uniform Korn
inequality is valid for a small enough thickness of the shell, and the
uniform constants depend in a bad way on the thickness. Consequently,
instabilities are to be expected in the numerical tests and special
approaches are necessary.
In the work [3], the above question is discussed in connection with optimal control problems governed by (elliptic) variational inequalities. In the case of convex control problems, it is known that they are equivalent with the first order system of optimality conditions. Therefore, the error estimates results valid for the finite element discretization of PDEs can be extended to optimal control problems. However, in nonconvex problems as described in the points (a), (b), (c) or in problems with constraints on the state, very little is known and high difficulties arise. The work [3] is a contribution in this respect and continues the research from [9].
References:
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[Contents] | [Index] |