Optimal design of membrane shells. Homogenization-based relaxation of the two-phase layout problem

The aim of using shell structures instead of plates is to avoid bending, hence the vital role of the membrane theory. Within this theory the classical optimum design problem is formulated: lay out two isotropic materials such that the shell becomes the stiffest possible. The amount of both the materials is fixed. The aim of the present paper is to reformulate this problem in a form assuring its well-posedness. The membrane approximation can be introduced from the very beginning or be imposed upon the relaxation. In the present paper it is shown that the latter modelling leads to a better formulation. It does not lose its stability even if one material degenerates to a void, thus leading to a well-posed shape design problem.