Shape design of shallow shells made of isotropic incompressible material

The present work is concerned with the subject of optimal design of elastic shells. The scope of the research includes shallow shells with the Mushtari-Donnell-Vlasov strain measures. We assume that the shell is made of one isotropic incompressible material, the amount of which is given. The research is based on the concept of so-called relaxed formulation of the minimum compliance problem, in which composite materials formed by fine mixing of the original material and voids are allowed in the analysis. The main goal of the optimization process is to distribute the composite within the shell body in a transversely homogeneous manner and in such a way that the external load work functional (i.e. compliance) achieves its minimal value. In order to find the optimal density of the material in the composite we make use of the alternate directions method introduced and developed by Allaire & Francfort (1993 ).