Optimal design of midsurface of shells: Differentiability proof and sensitivity computation

Abstract

Abstract. We suppose that a shell submitted to a given load (self-weight or wind, for instance), has to resist as well as possible towards given criteria. We aim at the following problem: Is it possible to find an optimal design of the midsurface of the shell with respect to this criteria?This problem can be worked using gradient-type algorithms. In this paper we work on the differentiability proof and numerical computation of the gradient.For a given shape of the midsurface, we consider that the shell works in linear elastic conditions. We use the Budiansky-Sanders model for elastic shells, from which we get the displacement field in the shell. The criteria to be minimized are supposed to depend on the shape directly, and also through the displacement field. In this paper, we prove that the displacement field depends on the shape in a Frrchet-differentiable manner (for an appropriate topology on the set of admissible shapes). Then we give a way to compute the gradient of a given criteria from a theoretical point of view and from a numerical point of view.This allows us to use descent-type methods of optimization. They will lead to shapes which react better and better. Notice that we know nothing about convergence of these methods, the existence and unicity of a theoretical optimal solution. But from a practical point of view, it is quite interesting to be able to modify a given shape to obtain a better one.

On the optimization of non periodic homogenized microstructures

“…More precisely, in order to use descent type methods we want to differentiate j( (p ) with respect to (p. We have the following classical result (see for instance [6] Vw e V. D…”

Shape optimization of an elastic thin shell under various criteria

“…(ii) For axisymmetric shells, Hlavacek (1983) and Mota Soares et al (1987) have studied the approximation of an optimization problem. (iii) The differentiability of u~ is also proved by Banichuk and Laxichev (1984) for shallow shells, by Chenais (1987) for the Budiansky-Sanders model (1967), and by Palma (1989) for the model due to Koiter (1966 and1970). …”

“…This methology was first applied to the mid line optimization of an arch (Chenais and Rousselet 1984) and extended to general shells by Chenais (1987) for the Budiansky-Sanders model (1987), and by Balma (1989) for the Koiter model (1966 and1970). For axisymmetric shells existence and approximation results may be found in Hlavace]~ (1983) whereas numerical results may be found in Mota Soares et al (1987).…”

Nash strategies for the inverse inclusion Cauchy-Stokes problem

“…To prove Proposition 3 above, let us first remark that we are in a classical elliptic case : the states (u φ 1 , p φ 1 ) and (u φ 2 , p φ 2 ) solve respectively problems (P 1 ε,β ) and (P 2 ε,β ) which are elliptic, and with a smooth dependence of energy functionals on φ, namely on the regularized parameter H ε,β (φ). It is then well known that the states (u φ 1 , p φ 1 ) and (u φ 2 , p φ 2 ) are Frechet-differentiable with respect to φ, see [40] or [20] page 107, the assumptions made therein are straightforward in our case. From other part, the assumption |∇φ| ≡0 is fulfilled by the reinitialisation step (17) which forces |∇φ| to be close to 1.…”