Reformulation of Optimal Design Problems

Abstract

Frequently, in structural optimization one may assign to the design parameters constraints that are independent of either the state function or the basic functional. For example, in the search for an optimal distribution of thickness h in a beam, plate, or shell we assign the constraint h ≥0, which is a direct consequence of the physical meaning of the specific design variable. In optimization problems in which the optimal distribution of thickness turns out to be 0 at some interior points in the domain of optimization Ω, this constraint may cause some difficulties. For this reason it is convenient to use a device based on an idea of Valentine, namely to introduce auxiliary functions that “automatically” take care of the assigned conditions. For example, if we introduce a new design variable φ related to h by the equality h =φ ², then obviously, the inequality h ≥0 is satisfied for arbitrary real values of φ.